Angles in Polygons

A polygon is a 2D shape with straight sides, such as a triangle, square, pentagon or hexagon.

The key to calculating angles in polygons is knowing how many sides the shape has and whether you need to find an interior angle, an exterior angle, or the sum of all the angles.

1. Learn the interior angle sum formula

The interior angles are the angles inside a polygon.
The formula for the sum of the interior angles is:

$$\boxed{{\displaystyle (n-2)\times {180}^{\circ }}}$$
Here, $n$ is the number of sides.

For example, a pentagon has 5 sides:

$$(5-2)\times {180}^{\circ }=3\times {180}^{\circ }=\boxed{{\displaystyle {540}^{\circ }}}$$
So, the five interior angles of a pentagon add up to ${540}^{\circ }$.

Interior angle sums to remember

Triangle — 3 sides, 180°

Quadrilateral — 4 sides, 360°

Pentagon — 5 sides, 540°

Hexagon — 6 sides, 720°

2. Calculate one interior angle in a regular polygon

A regular polygon has all sides equal and all interior angles equal.

To find one interior angle:

$$\boxed{{\displaystyle \frac{(n-2)\times {180}^{\circ }}{n}}}$$n(n−2)×180∘​​

Example: Find one interior angle of a regular hexagon.

Step 1: A hexagon has 6 sides, so $n=6$.

Step 2: Calculate the total interior angle sum.

$$(6-2)\times {180}^{\circ }={720}^{\circ }$$Step 3: Divide by the 6 equal angles.

$${720}^{\circ }÷6=\boxed{{\displaystyle {120}^{\circ }}}$$Each interior angle is ${120}^{\circ }$.

3. Calculate exterior angles

An exterior angle is formed by extending one side of the polygon. An interior angle and its adjacent exterior angle add up to ${180}^{\circ }$180∘.

$$\boxed{{\displaystyle \text{Exterior angle}={180}^{\circ }-\text{interior angle}}}$$For a regular polygon, you can also use:

$$\boxed{{\displaystyle \text{Exterior angle}=\frac{{360}^{\circ }}{n}}}$$For example, a regular pentagon has an exterior angle of:

$${360}^{\circ }÷5=\boxed{{\displaystyle {72}^{\circ }}}$$Check: ${180}^{\circ }-{108}^{\circ }={72}^{\circ }$, since each interior angle is ${108}^{\circ }$.

Remember: the exterior angles of any polygon add up to ${360}^{\circ }$ when you take one at each vertex in the same direction.

4. Find a missing angle

If a polygon is not regular, its angles might not all be equal. You can still find a missing angle by subtracting the known angles from the total.

Example: A quadrilateral has three angles of ${90}^{\circ }$, ${85}^{\circ }$ and ${110}^{\circ }$. Find the fourth angle.

Step 1: The interior angles of a quadrilateral add up to ${360}^{\circ }$360∘.

Step 2: Add the known angles.

$${90}^{\circ }+{85}^{\circ }+{110}^{\circ }={285}^{\circ }$$Step 3: Subtract from ${360}^{\circ }$360∘.

$${360}^{\circ }-{285}^{\circ }=\boxed{{\displaystyle {75}^{\circ }}}$$The missing angle is ${75}^{\circ }$.

Quick formula sheet

Exam tip: First count the sides, then identify which type of angle you need. That tells you which formula to use.

Angles in Polygons

1 / 10

The diagram shows a regular pentagon, ABCDE, and a square, EDFG.
The lines CD and DG are both sides of another regular polygon, P.
How many sides does polygon P have?

Check

2 / 10

The diagram shows three regular pentagons meeting at a point.
What is the size of the angle marked x?

Check

3 / 10

Shape A is a regular triangle. Shape B is a regular octagon.
Another regular polygon, P, is shown on the diagram
How many sides does polygon P have?

Check

4 / 10

ABCDEF is a hexagon.
Angle BAF = Angle ABC = Angle AFE = Angle BCD.
Angle DEF = Angle CDE = 130°.
What is the size of angle BAF?

Check

5 / 10

ABCDEF is a hexagon.
Angle CDE = 2 × Angle BCD
What is the size of angle CDE?

Check

6 / 10

ABCDE is a pentagon.
What is the size of angle ABC?

Check

7 / 10

The size of each interior angle in a regular polygon is 165°.
How many sides does the polygon have?

Check

8 / 10

The size of each exterior angle in a regular polygon is 18°.
How many sides does the polygon have?

Check

9 / 10

The size of each exterior angle in a regular polygon is 20°.
How many sides does the polygon have?

Check

10 / 10

What is the size of each interior angle in a regular hexagon?

Check

Your score is

The average score is 0%