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Angles of Polygons

In these lessons, we will learn

• how to calculate the sum of interior angles of a polygon using the sum of angles in a triangle
• the formula for the sum of interior angles in a polygon
• how to solve problems using the sum of interior angles
• the formula for the sum of exterior angles in a polygon
• how to solve problems using the sum of exterior angles.

All the polygons in these lessons are assumed to be convex polygons .

Related Pages Polygons Quadrilaterals Cyclic Quadrilaterals More Geometry Lessons

The following diagrams give the formulas for the sum of the interior angles of a polygon and the sum of exterior angles of a polygon. Scroll down the page if you need more examples and explanation.

Sum Of Interior Angles Of A Polygon

We first start with a triangle (which is a polygon with the fewest number of sides). We know that

The sum of interior angles in a triangle is 180°.

This is also called the Triangle Sum Theorem. Click here if you need a proof of the Triangle Sum Theorem.

Next, we can figure out the sum of interior angles of any polygon by dividing the polygon into triangles. We can separate a polygon into triangles by drawing all the diagonals that can be drawn from one single vertex.

In the quadrilateral shown below, we can draw only one diagonal from vertex A to vertex B. So, a quadrilateral can be separated into two triangles.

The sum of angles in a triangle is 180°. Since a quadrilateral is made up of two triangles the sum of its angles would be 180° × 2 = 360°

The sum of interior angles in a quadrilateral is 360º

A pentagon (five-sided polygon) can be divided into three triangles. The sum of its angles will be 180° × 3 = 540°

The sum of interior angles in a pentagon is 540°.

A hexagon (six-sided polygon) can be divided into four triangles. The sum of its angles will be 180° × 4 = 720°

The sum of interior angles in a hexagon is 720°.

Formula For The Sum Of Interior Angles

We can see from the above examples that the number of triangles in a polygon is always two less than the number of sides of the polygon. We can then generalize the results for a n-sided polygon to get a formula to find the sum of the interior angles of any polygon.

The following diagram shows the formula for the sum of interior angles of an n-sided polygon and the size of an interior angle of a n-sided regular polygon. Scroll down the page for more examples and solutions on the interior angles of a polygon.

Example: Find the sum of the interior angles of a heptagon (7-sided)

Solution: Step 1: Write down the formula (n - 2) × 180°

Step 2: Plug in the values to get (7 - 2) × 180° = 5 × 180° = 900°

Answer: The sum of the interior angles of a heptagon (7-sided) is 900°.

Example: Find the interior angle of a regular octagon.

Answer: Each interior angle of an octagon (8-sided) is 135°.

Worksheet using the Formula for the Sum of Interior Angles

How to find the sum of the interior angles of any polygon using triangles and then derive the generalized formula?

Problems Using The Sum Of Interior Angles

How to find a missing angle using the sum of interior angles of a polygon?

How to use the sum of interior angles to write an equation and solve for the unknown? Write an equation and solve for the unknown. Substitute your answer into each expression to determine the measure of the angle. Give reasons for your answers.

Formula For The Sum Of Exterior Angles

The sum of exterior angles of any polygon is 360°.

The exterior angle of a regular n-sided polygon is 360°/n

Worksheet using the formula for the sum of exterior angles

Worksheet using the formula for the sum of interior and exterior angles

How to find the sum of the exterior angles and interior angles of a polygon? Every convex polygon has interior and exterior angles. The interior angles are inside the polygon formed by the sides. The exterior angles form a linear pair with the interior angles.

Example: Determine the measure of each exterior and interior angle of a regular polygon.

Problems using the sum of exterior angles

The following video shows a problem involving the sum of exterior angles of a polygon.

Example: A regular polygon has an exterior angle that measures 40°. How many sides does the polygon have?

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Unit 2: Angles

About this unit.

In this topic, we will learn what an angle is and how to label, measure and construct them. We will also explore special types of angles.

Angle introduction

• Angles: introduction (Opens a modal)
• Naming angles (Opens a modal)
• Angle basics review (Opens a modal)
• Angle basics 4 questions Practice
• Name angles 4 questions Practice

Measuring angles

• Measuring angles in degrees (Opens a modal)
• Measuring angles using a protractor (Opens a modal)
• Measuring angles using a protractor 2 (Opens a modal)
• Measuring angles review (Opens a modal)
• Measure angles 4 questions Practice

Constructing angles

• Constructing angles (Opens a modal)
• Constructing angles review (Opens a modal)
• Draw angles 7 questions Practice

Angles in circles

• Angle measurement & circle arcs (Opens a modal)
• Angles in circles word problems (Opens a modal)
• Angles in circles 7 questions Practice

Angle types

• Recognizing angles (Opens a modal)
• Drawing acute, right and obtuse angles (Opens a modal)
• Identifying an angle (Opens a modal)
• Angle types review (Opens a modal)
• Angle types 4 questions Practice
• Recognize angles in figures 4 questions Practice
• Draw right, acute, and obtuse angles 7 questions Practice
• Benchmark angles 7 questions Practice

Vertical, complementary, and supplementary angles

• Complementary & supplementary angles (Opens a modal)
• Complementary and supplementary angles review (Opens a modal)
• Vertical angles review (Opens a modal)
• Angle relationships example (Opens a modal)
• Vertical angles are congruent proof (Opens a modal)
• Identifying supplementary, complementary, and vertical angles 7 questions Practice
• Complementary and supplementary angles (visual) 4 questions Practice
• Complementary and supplementary angles (no visual) 7 questions Practice
• Vertical angles 4 questions Practice

Angles between intersecting lines

• Angles, parallel lines, & transversals (Opens a modal)
• Parallel & perpendicular lines (Opens a modal)
• Missing angles with a transversal (Opens a modal)
• Parallel lines & corresponding angles proof (Opens a modal)
• Missing angles (CA geometry) (Opens a modal)
• Proving angles are congruent (Opens a modal)
• Proofs with transformations (Opens a modal)
• Angle relationships with parallel lines 7 questions Practice
• Line and angle proofs 4 questions Practice

Sal's old angle videos

• Intro to angles (old) (Opens a modal)
• Angles (part 2) (Opens a modal)
• Angles (part 3) (Opens a modal)
• Angles formed between transversals and parallel lines (Opens a modal)
• Angles of parallel lines 2 (Opens a modal)
• The angle game (Opens a modal)
• The angle game (part 2) (Opens a modal)
• Acute, right, & obtuse angles (Opens a modal)

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Interior Angles Of A Polygon

Here we will learn about interior angles in polygons including how to calculate the sum of interior angles for a polygon, single interior angles and use this knowledge to solve problems. 

There are also angles in polygons worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are interior angles?

Interior angles are the angles inside a shape. They are the angles within a polygon made by two sides:

Interior and exterior angles form a straight line – they add to 180° :

We can calculate the sum of the interior angles of a polygon by splitting it into triangles and multiplying the number of triangles by 180° .

The number of triangles a polygon can be split into is always 2 less than the number of sides.

A heptagon has 7 sides.

7 – 2 = 5 , so we can split the heptagon into 5 triangles:

The general formula is:

Sum of Interior Angles = (n – 2) × 180° ‘ n ’ is the number of sides the polygon has

Step by step guide: Angles in polygons  

What are interior angles of polygons?

• Polygon : A polygon is a two dimensional shape with at least three sides, where the sides are all straight lines. 
• Regular & irregular polygons : A regular polygon is where all angles are equal size and all sides are equal length   E.g. a square An irregular polygon is where all angles are not equal size and/or all sides are not equal length E.g. a trapezium.

How to solve problems involving interior angles

In order to solve problems involving interior angles:

• Identify the number of sides in any polygon/s given in the question. Note whether the shape is regular or irregular.

Find the sum of interior angles for any polygon/s given.

• Identify what the question is asking.

Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of exterior angles \, \textbf{=} \, \bf{360^{\circ}} if required

How to solve problems involving interior angles of a polygon.

Interior and exterior angles worksheet (includes interior angles of a polygon)

Get your free interior and exterior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on angles in polygons

Interior angles of a polygon  is part of our series of lessons to support revision on  angles in polygons . You may find it helpful to start with the main angles in polygons lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Angles in polygons
• Exterior angles of a polygon
• Angles in a triangle
• Angles in a quadrilateral
• Angles in a pentagon
• Angles in a hexagon

Interior angles examples

Example 1: finding a single interior angle of a regular polygon.

Find the size of each interior angle for a regular decagon.

• Identify the number of sides in any polygon/s given in the question. Note whether this are regular or irregular shapes.

10 sides – regular shape.

2 Find the sum of interior angles for any polygon/s given.

Sum of interior angles = (n – 2) × 180°

As a decagon has 10 sides:

n=10 , so we can substitute n=10 into the formula.

Sum of interior angles of a decagon = (10 – 2) × 180°

Sum of  interior angles of a decagon = 8 × 180°

Sum of  interior angles of a decagon = 1440°

3 Identify what the question is asking you to find.

The question is asking for ‘each interior angle’. This means the size of one interior angle.

4 Solve the problem using the information you have already gathered with use of the formulae interior angle \textbf{+} exterior angle \, \textbf{=} \; \bf{180^{\circ}} and Sum of exterior angles \, \textbf{=} \, \bf{360^{\circ}} if required

We know the sum of the interior angles for this polygon is 1440° .

We know, as it is a regular polygon, that all the angles are of equal size.

Therefore we can find the size of each interior angle by dividing the sum of interior angles by the number of angles in the polygon:

The size of each interior angle is 144° .

Step-by-step guide: Substitution

Example 2: finding a single interior angle of an irregular polygon

The diagram shows a polygon. Find the size of angle x .

Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular shapes.

6 sides – irregular hexagon

Sum of interior angles  = (n – 2) × 180°

Sum of interior angles for a hexagon = (6 – 2) × 180°

Sum of interior angles for a hexagon = 720°

Identify what the question is asking you to find.

Finding the missing angle labelled as x .

Note that we know the values of all the other angles.

The size of angle is 119° .

Example 3: finding the number of sides given the interior angle of a regular polygon 

Each of the interior angles of a regular polygon is 140° . How many sides does the polygon have?

Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular shapes.

Unknown number of sides – regular shape

We need to find the number of sides.

We know a single angle of this regular polygon is 140° .

Therefore all the angles are 140° .

We can write the sum of the interior angles as 140 multiplied by the number of sides or 140n .

The polygon has 9 sides.

Note: We can also solve this problem by calculating an exterior angle.

Example 4: multiple shapes

Shown below are three congruent regular pentagons. Find angle y .

Each polygon has 5 sides (pentagon) and is regular.

As each polygon shown is a regular pentagon they all have equal sums of their interior angles:

Sum of interior angles for a pentagon = (5 – 2) × 180°

Sum of interior angles for each pentagon = 540°

Find the missing angle y shown on the diagram.

We know that angles around a point add to 360° , so if we add the three interior angles shown and y together we will get 360° .

Each interior angle shown is 540 ÷ 5 = 108°

We can now calculate y by forming an equation:

Angle y is equal to 36° .

Example 5: problem solving to find the number of sides

Shown below are sections of three identical regular polygons where AB, BC and CA are all sides of the polygons. 

ABC is an equilateral triangle formed by placing the three larger polygons together.

Calculate the number of sides each regular polygon has.

Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular shapes .

Shown is an equilateral triangle (regular shape) made up of the adjacent sides AB, BC and CA .

We need to calculate the number of sides of the larger polygons.

An equilateral triangles has the sum of interior angles of 180° .

We do not know the number of sides of the polygons so their sum of interior angles can be represented by (n – 2) × 180° .

The number of sides of the regular polygons where we are only shown one side. 

Looking at point A we can see there are three angles around a point. One of the angles is within the equilateral triangle, so it must be 60° , and the other two angles are from the polygons we are attempting to find. 

We will call these angles x :

We know that angles around a point add to 360° .

This means that each interior angle of the regular polygon is 150° .

So the sum of interior angles is equal to 150 × n or 150n : 

150n = (n – 2) × 180

We can now solve for n :

The polygon has 12 sides, so each polygon shown in the diagram has 1 2 sides.

Example 6: problem solving to find the number of angles

Shown is a regular pentagon. Find y .

5 sides – regular

Sum of interior angles for a decagon = (5 – 2) × 180°

Sum of interior angles for a decagon = 540°

Find angle y which is within one of the interior angles.

As the polygon is regular you can find the size of one interior angle by:

540° ÷ 5 = 108

As the polygon is regular AC = AB

Therefore ABC is an isosceles triangle where angles ACB and ABC are equal to one another and are therefore both y .

We know that the interior angles of a triangle add to 180° . 

Common misconceptions

• Miscounting the number of sides
• Misidentifying if a polygon is regular or irregular 
• Dividing the sum of interior angles by the number of triangles created. You should divide by the number of sides to find the size of one interior angle (for regular polygons only)
• Incorrectly assuming all the angles are the same size
• Misidentifying which angle the questions is asking you to calculate

Practice interior angles of a polygon questions

1. Find the sum of interior angles for a polygon with 13 sides

Sum of Interior Angles = (n-2)\times180

In this case n=13 , so the calculation becomes 11 \times 180 .

2. Find the size of one interior angle for a regular quadrilateral

The sum of interior angles in a quadrilateral is 360^{\circ} . For a regular shape all the angles are the same size, so we divide 360 by 4 to arrive at the answer.

3. Find the size of one interior angle for a regular nonagon

The sum of interior angles in a nonagon is 1260^{\circ} . For a regular shape all the angles are the same size, so we divide 1260 by 9 to arrive at the answer.

4. Each of the interior angles of a regular polygon is 165^{\circ} . How many sides does the polygon have?

With this in mind, we have 165n=(n-2) \times 180

Which simplifies to 15n = 360

5. Each of the interior angles of a regular polygon is 160^{\circ} . How many sides does the polygon have?

With this in mind, we have 160n=(n-2)\times180

Which simplifies to 20n = 360

6. Four interior angles in a pentagon are each 115^{\circ} . Find the size of the other angle.

By using the formula,

We know that a pentagon has interior angles that add up to 540^{\circ} .

540 – (4 \times 155) = 80

Interior angles of a polygon gcse questions.

1. Work out the size of the angle labeled x .

(6-2) \times 180 = 720

80 + 55 + 280 + 25 + 162 = 602

720-602=118

2. The diagram below shows a regular decagon.

(a) Work out the size of angle a .

(b) Work out the size of angle b .

(10-2) \times 180 = 1440

1440 \div 10 = 144

144\times 2 = 288

360 – 288 = 72

72 \div 2=36

3. A regular polygon’s interior and exterior angles are in the ratio 9 : 1 . How many sides does the polygon have?

180^{\circ} in ratio 9 : 1

180 \div 10=18, 18 \times 9=162, 18 \times 1= 18

One interior angle = 162^{\circ}

\begin{aligned} 162n&=(n-2) \times 180 \\ 162n&=180n-360 \end{aligned}

Learning checklist

You have now learned how to:

• Use conventional terms for geometry e.g. interior angle
• Knowing names and properties of polygons
• Calculate the sum of interior angles for a regular polygon
• Derive and use the sum of angles in a triangle to deduce and use the angle sum in any polygon, and to derive properties of regular polygons
• Calculate the size of the interior angle of a regular polygon

The next lessons are

• Angle rules
• Angles in parallel lines
• How to calculate volume

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General Polygons - Problem Solving

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Regular \(n\)-gon

We shall first study regular polygons. A regular polygon is a polygon in which all sides have the same length and all angles are equal in measure. A rhombus is not a regular polygon, though all sides are equal. This is because all angles are not equal. Let \(S\) be the side length of an \(n\)-sided polygon (\(n\)-gon), \(R\) the circumradius, and \(r\) the inradius.

An \(n\)-gon is made up of \(n\) isosceles triangles with base \(S,\) sides \(R,\) and angle between the two sides measuring \(\frac{2\pi} n.\)

For relations between various lengths and angles, the right-angled triangle, which is half of the isosceles triangle, is helpful. So we get the relation \[S=2\times R\sin\frac \pi n ~\text{ or }~ R=\dfrac S{2\times \sin\frac \pi n }\] and \[S=2\times r\tan\frac \pi n ~\text{ or }~ r=\dfrac {S\times \cot\frac \pi n } 2. \] The base angles are \(\frac{(n-2)\times \pi}{2n},\) while the angle between adjacent sides is \(\frac{(n-2)\times\pi}{n}.\)

The sketch below shows how the isosceles triangles are placed with a common vertex, coinciding sides, and the angles. Apart from the solution of the Isosceles triangle, the bases of \(n\)-gon solution, the lengths of diagonals \(D_2, D_3, D_4, \ldots\) are important.

For an \(n\)-gon, there will be \(n\) vertices \(A_0, A_1, \ldots, A_{n-2}, A_{n-1}. \) Then \((n-1)\) lines can be drawn from \(A_0\) to othere vertices. Out of these, \(A_0A_1\) and \(A_0A_{n-1} \) are adjacent sides. So there are only \((n-3)\) diagonals. If \(n\) is even, there will be one big diagonal, and all the \((n-2)\) remaining ones will be in \(\frac{n-4} 2\) pairs, as shown in the sketch. If \(n\) is odd, all \((n-3)\) diagonals will be in \(\frac{n-3} 2\) pairs. What is true for vertex \(A_0\) is true for each of the \(n\) vertices.

Next, we have to find out the lengths of diagonals.

The diagonal \(D_n\) between \(A_0A_k\) also forms an isosceles triangle with sides \(R\) and the angle \(\alpha=\frac{2\pi} n\times k.\) Now we can solve these as we had solved for sides.

A regular hexagon swimming pool has a side length of \(3\) meters. It is surrounded by a sidewalk of uniform width \(1.2\) meters. The sidewalk is surrounded by a grass landscape of uniform width \(0.4\) meters.

Which of the following is the combined area of the sidewalk and landscape?

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Polygons: Formula and Examples

Exterior angles and interior angles, interior angle sum theorem, what is true about the sum of interior angles of a polygon .

The sum of the measures of the interior angles of a convex polygon with n sides is $ (n-2)\cdot180^{\circ} $

What is the total number degrees of all interior angles of a triangle ?

You can also use Interior Angle Theorem :$$ (\red 3 -2) \cdot 180^{\circ} = (1) \cdot 180^{\circ}= 180 ^{\circ} $$

What is the total number of degrees of all interior angles of the polygon ?

360° since this polygon is really just two triangles and each triangle has 180°

You can also use Interior Angle Theorem :$$ (\red 4 -2) \cdot 180^{\circ} = (2) \cdot 180^{\circ}= 360 ^{\circ} $$

What is the sum measure of the interior angles of the polygon (a pentagon) ?

Use Interior Angle Theorem :$$ (\red 5 -2) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ} $$

What is sum of the measures of the interior angles of the polygon (a hexagon) ?

Video Tutorial on Interior Angles of a Polygon

Definition of a Regular Polygon:

Examples of regular polygons.

Measure of a Single Interior Angle

What about when you just want 1 interior angle.

In order to find the measure of a single interior angle of a regular polygon  (a polygon with sides of equal length and angles of equal measure) with n sides, we calculate the sum interior angles or $$ (\red n-2) \cdot 180 $$ and then divide that sum by the number of sides or $$ \red n$$.

$ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} $

So, our new formula for finding the measure of an angle in a regular polygon is consistent with the rules for angles of triangles that we have known from past lessons.

To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: $ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} \\ \frac{(\red8-2) \cdot 180}{ \red 8} = 135^{\circ} $

Finding 1 interior angle of a regular Polygon

What is the measure of 1 interior angle of a regular octagon?

Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle

Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle

Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)?

Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle

Challenge Problem

What is the measure of 1 interior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. You can only use the formula to find a single interior angle if the polygon is regular!

Consider, for instance, the ir regular pentagon below.

You can tell, just by looking at the picture, that $$ \angle A    and    \angle B $$ are not congruent .

The moral of this story- While you can use our formula to find the sum of the interior angles of any polygon (regular or not), you can not use this page's formula for a single angle measure--except when the polygon is regular .

How about the measure of an exterior angle?

Formula for sum of exterior angles: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.

Measure of a Single Exterior Angle

$$ \angle1 + \angle2 + \angle3 = 360° $$

$$ \angle1 + \angle2 + \angle3 + \angle4 = 360° $$

$$ \angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360° $$

Practice Problems

Calculate the measure of 1 exterior angle of a regular pentagon?

Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?

Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. Although you know that sum of the exterior angles is 360 , you can only use formula to find a single exterior angle if the polygon is regular!

Consider, for instance, the pentagon pictured below. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$ \angle A \text{ and } and \angle B $$ are not congruent..

Determine Number of Sides from Angles

It's possible to figure out how many sides a polygon has based on how many degrees are in its exterior or interior angles.

If each exterior angle measures 10°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

If each exterior angle measures 20°, how many sides does this polygon have?

If each exterior angle measures 15°, how many sides does this polygon have?

If each exterior angle measures 80°, how many sides does this polygon have?

When you use formula to find a single exterior angle to solve for the number of sides , you get a decimal (4.5), which is impossible. Think about it: How could a polygon have 4.5 sides? A quadrilateral has 4 sides. A pentagon has 5 sides.

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Interior angle of polygons

Exterior angle of polygons, practice questions, angles in polygons – explanation & examples.

How to differentiate them then? ANGLES!

The simplest example is that both rectangle and a parallelogram have 4 sides each, with opposite sides are parallel and equal in length. The difference lies in angles, where a rectangle has 90-degree angles on its all 4 sides while a parallelogram has opposite angles of equal measure.

In this article, you will learn:

• How to find the angle of a polygon?
• Interior angles of a polygon.
• Exterior angles of a polygon.
• How to calculate the size of each interior and exterior angle of a regular polygon.  

How to Find the Angles of a Polygon?

We know that a polygon is a two-dimensional multi-sided figure made up of straight-line segments . The sum of angles of a polygon is the total measure of all interior angles of a polygon.

Since all the angles inside the polygons are the same. Therefore, the formula for finding the angles of a regular polygon is given by;

Sum of interior angles = 180° * (n – 2)

Where n = the number of sides of a polygon.

• Angles of a Triangle:

a triangle has 3 sides, therefore,

Substitute n = 3 into the formula of finding the angles of a polygon.

= 180° * (3 – 2)

• Angles of a Quadrilateral:

A quadrilateral is a 4-sided polygon, therefore,

By substitution,

sum of angles = 180° * (n – 2)

= 180° * (4 – 2)

• Angles of a Pentagon

A pentagon is a 5 – sided polygon.

Substitute.

=180° * (5 – 2)

• Angles of an octagon.

An Octagon is an 8 – sided polygon

= 180° * (8 – 2)

Angles of a Hectagon:

a Hectagon is a 100-sided polygon.

= 180° * (100 – 2)

= 180° * 98

The interior angle is an angle formed inside a polygon, and it is between two sides of a polygon.

The number of sides in a polygon is equal to the number of angles formed in a particular polygon. The size of each interior angle of a polygon is given by;

Measure of each interior angle = 180° * (n – 2)/n

where n = number of sides.

• Size of the interior angle of a decagon.

A decagon is a 10 -sided polygon.

Substitution.

= 180° * (10 – 2)/10

= 180° * 8/10

• Interior angle of a Hexagon.

A hexagon has 6 sides. Therefore, n = 6

Measure of each interior angle =180° * (n – 2)/n

= 180° * (6 – 2)/6

= 180° * 4/6

• Interior angle of a rectangle

A rectangle is an example of a quadrilateral (4 sides)

=180° * (4 – 2)/4

=180° * 1/2

• Interior angle of a pentagon.

A pentagon is composed of 5 sides.

The measure of each interior angle =180° * (5 – 2)/5

=180° * 3/5

The exterior angle is the angle formed outside a polygon between one side and an extended side. The measure of each exterior angle of a regular polygon is given by;

The measure of each exterior angle =360°/n, where n = number of sides of a polygon.

• Exterior angle of a triangle:

For a triangle, n = 3

Measure of each exterior angle = 360°/n

• Exterior angle of a Pentagon:

NOTE: The interior angle and exterior angle formulas only work for regular polygons. Irregular polygons have different interior and exterior measures of angles.

Let’s look at more example problems about interior and exterior angles of polygons.

The interior angles of an irregular 6-sided polygon are; 80°, 130°, 102°, 36°, x°, and 146°.

Calculate the size of angle x in the polygon.

For a polygon with 6 sides, n = 6

the sum of interior angles =180° * (n – 2)

= 180° * (6 – 2)

Therefore, 80° + 130° + 102° +36°+ x° + 146° = 720°

494° + x = 720°

Subtract 494° from both sides.

494° – 494° + x = 720° – 494°

Find the exterior angle of a regular polygon with 11 sides.

The measure of each exterior angle= 360°/n

The exterior angles of a polygon are; 7x°, 5x°, x°, 4x° and x°. Determine the value of x.

Sum of exterior =360°

7x° + 5x° + x° + 4x° + x° =360°

Divide both sides by 18.

x = 360°/18

Therefore, the value of x is 20°.

What is the name of a polygon whose interior angles are each 140°?

Size of each interior angle = 180° * (n – 2)/n

Therefore, 140° = 180° * (n – 2)/n

Multiply both sides by n

140°n =180° (n – 2)

140°n = 180°n – 360°

Subtract both sides by 180°n.

140°n – 180°n = 180°n – 180°n – 360°

-40°n = -360°

Divide both sides by -40°

n = -360°/-40°

Therefore, the number of sides is 9 (nonagon).

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Appendix B: Geometry

Using properties of angles to solve problems, learning outcomes.

• Find the supplement of an angle
• Find the complement of an angle

Are you familiar with the phrase ‘do a [latex]180[/latex]?’ It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is [latex]180[/latex] degrees. See the image below.

[latex]\angle A[/latex] is the angle with vertex at [latex]\text{point }A[/latex].

We measure angles in degrees, and use the symbol [latex]^ \circ[/latex] to represent degrees. We use the abbreviation [latex]m[/latex] to for the measure of an angle. So if [latex]\angle A[/latex] is [latex]\text{27}^ \circ [/latex], we would write [latex]m\angle A=27[/latex].

If the sum of the measures of two angles is [latex]\text{180}^ \circ[/latex], then they are called supplementary angles. In the images below, each pair of angles is supplementary because their measures add to [latex]\text{180}^ \circ [/latex]. Each angle is the supplement of the other.

The sum of the measures of supplementary angles is [latex]\text{180}^ \circ [/latex].

The sum of the measures of complementary angles is [latex]\text{90}^ \circ[/latex].

Supplementary and Complementary Angles

If the sum of the measures of two angles is [latex]\text{180}^\circ [/latex], then the angles are supplementary .

If angle [latex]A[/latex] and angle [latex]B[/latex] are supplementary, then [latex]m\angle{A}+m\angle{B}=180^\circ[/latex].

If the sum of the measures of two angles is [latex]\text{90}^\circ[/latex], then the angles are complementary .

If angle [latex]A[/latex] and angle [latex]B[/latex] are complementary, then [latex]m\angle{A}+m\angle{B}=90^\circ[/latex].

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.

Use a Problem Solving Strategy for Geometry Applications.

• Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
• Identify what you are looking for.
• Name what you are looking for and choose a variable to represent it.
• Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
• Solve the equation using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Answer the question with a complete sentence.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

An angle measures [latex]\text{40}^ \circ[/latex].

1. Find its supplement

2. Find its complement

Write the appropriate formula for the situation and substitute in the given information. [latex]m\angle A+m\angle B=90[/latex] Step 5. Solve the equation. [latex]c+40=90[/latex]

[latex]c=50[/latex] Step 6. Check:

[latex]50+40\stackrel{?}{=}90[/latex]

In the following video we show more examples of how to find the supplement and complement of an angle.

Did you notice that the words complementary and supplementary are in alphabetical order just like [latex]90[/latex] and [latex]180[/latex] are in numerical order?

Two angles are supplementary. The larger angle is [latex]\text{30}^ \circ[/latex] more than the smaller angle. Find the measure of both angles.

• Question ID 146497, 146496, 146495. Authored by : Lumen Learning. License : CC BY: Attribution
• Determine the Complement and Supplement of a Given Angle. Authored by : James Sousa (mathispower4u.com). Located at : https://youtu.be/ZQ_L3yJOfqM . License : CC BY: Attribution
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Angles, lines and polygons - Edexcel Polygons

Polygons are multi-sided shapes with different properties. Shapes have symmetrical properties and some can tessellate.

Part of Maths Geometry and measure

A polygon close polygon A polygon is a 2-dimensional closed shape with straight sides, eg triangle, hexagon, etc. is a 2D close two-dimensional (2D) Having only two dimensions, usually length (or height) and width. shape with at least three sides.

Types of polygon

Polygons can be regular or irregular. If the angles are all equal and all the sides are equal length it is a regular polygon.

Interior angles of polygons

To find the sum of interior angles in a polygon divide the polygon into triangles.

The sum of interior angles in a triangle is 180°. To find the sum of interior angles of a polygon, multiply the number of triangles in the polygon by 180°.

Calculate the sum of interior angles in a pentagon.

A pentagon contains 3 triangles. The sum of the interior angles is:

\(180 \times 3 = 540^\circ\)

The number of triangles in each polygon is two less than the number of sides.

The formula for calculating the sum of interior angles is:

\((n - 2) \times 180^\circ\) (where \(n\) is the number of sides)

Calculate the sum of interior angles in an octagon.

Show answer Hide answer

Using \((n - 2) \times 180^\circ\) where \(n\) is the number of sides:

\((8 - 2) \times 180 = 1,080^\circ\)

Calculating the interior angles of regular polygons

All the interior angles in a regular polygon are equal. The formula for calculating the size of an interior angle is:

\(\text{interior angle of a polygon} = \text{sum of interior angles} \div \text{number of sides}\)

Calculate the size of the interior angle of a regular hexagon close hexagon A polygon with six sides. .

The sum of interior angles is \((6 - 2) \times 180 = 720^\circ\).

One interior angle is \(720 \div 6 = 120^\circ\).

Exterior angles of polygons

If the side of a polygon is extended, the angle formed outside the polygon is the exterior angle.

The sum of the exterior angles of a polygon is 360°.

Calculating the exterior angles of regular polygons

The formula for calculating the size of an exterior angle is:

\(\text{exterior angle of a polygon} = 360 \div \text{number of sides}\)

Remember the interior and exterior angle add up to 180°.

Calculate the size of the exterior and interior angle in a regular pentagon close pentagon A polygon with five sides. .

The sum of exterior angles is 360°.

The exterior angle is \(360 \div 5 = 72^\circ\).

The interior and exterior angles add up to 180°.

The interior angle is \(180 - 72 = 108^\circ\).

The sum of interior angles is \((5 - 2) \times 180 = 540^\circ\).

The interior angle is \(540 \div 5 = 108^\circ\).

The exterior angle is \(180 - 108 = 72^\circ\).

More guides on this topic

• Loci and constructions - Edexcel
• 2-dimensional shapes - Edexcel
• 3-dimensional shapes - Edexcel
• Circles, sectors and arcs - Edexcel
• Circle theorems - Higher - Edexcel
• Transformations - Edexcel
• Pythagoras' theorem - Edexcel
• Units of measure - Edexcel
• Trigonometry - Edexcel
• Vectors - Edexcel

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Angles in Polygons (Challenges – Part 1)

Subject: Mathematics

Age range: 11-14

Resource type: Lesson (complete)

Last updated

17 January 2019

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Angles and Polygons

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Interior and Exterior Angles of Regular Polygons Word Problems

The angles that lie inside a shape, generally a polygon, are said to be interior angles .

An exterior angle of a polygon is the angle that is formed between any side of the polygon and a line extended from the next side. Every polygon has interior and exterior angles. The exterior is the term opposite to the interior which means outside. Therefore exterior angles can be found outside the polygon. The sum of the exterior angles of any polygon is equal to 360°. Any flat shape or figure is said to have interior or exterior angles only if it is a closed shape.

To find the sum of all interior angles in a regular polygon: The sum of the interior angles of a polygon can be found by taking the number of sides (n) and subtracting 2. Then, multiply that number by 180.

Sum of interior angles = (n – 2) ∙ 180°

To find the measure of each interior angle in a regular polygon: 1. To find the measure of one interior angle in a regular polygon, first find the sum of interior angles of the required polygon using the formula given below. Sum of interior angles = (n – 2) ∙ 180°

2. Next, divide the sum of interior angles by the total number of angles the regular polygon has.

To find the measure of each exterior angle in a regular polygon: The measure of one of the exterior angles of a regular polygon can be found by dividing 360 degrees by the number of angles (n).

Bees build honeycombs with hexagonal cells. What is the measure of each interior angle of the cell?

To find the measure of one interior angle in a regular polygon, first find the sum of interior angles of the required polygon using the formula given below.

A hexagon has 6 sides, so: (6 – 2) ∙ 180 = 4 ∙ 180 = 720°

Since this is a regular hexagon, all of the angles are equal, so divide the sum of the interior angles by 6.

Practice Interior and Exterior Angles of Regular Polygons Word Problems

Practice Problem 1

Practice Problem 2

Practice Problem 3

Polygon – A closed figure formed by three or more segments called sides.​

Interior angle – An angle of a polygon formed by two of its side and is inside the polygon. ​​

Exterior angle – An angle formed by one side and the extension of the adjacent side. It is outside the polygon.

Pre-requisite Skills Classify Angles Drawing Angles Estimating Angles Angle Relationships Classify Triangles Angles Finding Angle Measures Complementary and Supplementary Angles Angles in Triangles

Related Skill Geometric Proof