Book A Free Math Class

# Angles in a Pentagon: Interior, Exterior, and How to Find Them

[#Geometry](/content/tag/geometry/index.html)

TL;DR

The interior angles of any pentagon always add to 540°, found with the formula (n − 2) × 180° where n = 5. In a regular pentagon each interior angle is 108° and each exterior angle is 72°. This article shows where 540° comes from, how to find a missing angle in an irregular pentagon, and the slips students make along the way.

## What Are The Angles In A Pentagon?

A **pentagon** is a closed two-dimensional shape with **five straight sides and five vertices**. The **angles in a pentagon** are the five angles formed inside it where two sides meet (the **interior angles**) and the five angles formed outside when a side is extended (the **exterior angles**). For any pentagon, the **sum of the interior angles is 540°**. For a **regular pentagon** — one with five equal sides and five equal angles — each interior angle measures **108°**.

Read that twice, because the two numbers do different jobs. The 540° is a total that holds for _every_ pentagon. The 108° only applies when the pentagon is regular and you divide that total evenly across five equal corners.

## Where The 540° Comes From

You do not have to memorise 540°. You can rebuild it from a triangle, which is the move worth carrying with you.

Pick any one vertex of the pentagon and draw straight lines from it to the other non-adjacent vertices. A pentagon splits cleanly into **three triangles** this way. Every triangle's interior angles add to 180° (the [triangle sum theorem](/content/math/geometry/triangle-sum-theorem/index.html)). Three triangles means three lots of 180°.

Sum of interior angles=3×180°=540°

That triangle-counting idea generalises into one formula for any polygon with n sides:

Sum of interior angles=(n−2)×180°

**Variable glossary:** n is the number of sides; (n − 2) is the number of triangles the polygon splits into; 180° is the angle sum of one triangle.

For a pentagon, n = 5:

(5−2)×180°=3×180°=540°

To get **one** interior angle of a _regular_ pentagon, share the total across five equal corners:

Each interior angle=540°5=108°

The **exterior angle** of a regular pentagon is what is left when you turn the corner. Interior and exterior angles at the same vertex sit on a straight line, so they add to 180°:

Each exterior angle=180°−108°=72°

And the exterior angles of _any_ polygon always add to 360°, which gives the same answer a second way: 360° ÷ 5 = 72°. If you ever want the longer treatment, the [exterior angles of a polygon](/content/math/geometry/exterior-angles-of-a-polygon/index.html) page walks through why that 360° total never changes.

## Interior, Exterior, And Central Angles Side By Side

These three angle names get mixed up constantly, so here they are next to each other.

| Angle type            | What it is                                                  | Regular pentagon value |  
| --------------------- | ---------------------------------------------------------- | ----------------------- |  
| Interior angle        | Angle inside, where two sides meet                         | 108°                   |  
| Exterior angle        | Angle between a side and the extension of the next side    | 72°                    |  
| Central angle         | Angle at the center, between lines to two adjacent vertices | 72°                    |  
| Sum of interior angles | Total of all five interior angles                          | 540°                   |

The exterior and central angles happen to match at 72° in a regular pentagon. That is a coincidence of the regular case, not a rule — do not carry it over to irregular pentagons.

## How Do You Find A Missing Angle In An Irregular Pentagon?

This is the question that actually shows up on homework, and it is the strongest reason the 540° total matters. The pentagons in textbooks are rarely regular. They are irregular — different sides, different angles — and you are given four of the five angles and asked for the fifth. The trick is that the total is still locked at 540°.

## Examples of Angles in a Pentagon

### Example 1

**Find the sum of the interior angles of a pentagon.**

Use the polygon formula with n = 5.

Sum = (n − 2) × 180°

Sum = (5 − 2) × 180°

Sum = 3 × 180°

**Final answer:** 540°

### Example 2

**A student says each interior angle of a regular pentagon is 540° ÷ 4 = 135°. Find the correct value.**

Here is the wrong path first, because it is the single most common error.

Wrong attempt: the student remembered to divide the 540° total but divided by 4 instead of 5, perhaps thinking of the three triangles plus something, or just miscounting corners.

540° ÷ 4 = 135°

That answer is too big. A regular pentagon clearly has corners that look less than a right-angle-and-a-half, and 135° is the interior angle of a regular _octagon_, not a pentagon. The break is the divisor.

Correct method: a pentagon has five equal corners, so divide by 5.

540° ÷ 5 = 108°

**Final answer:** 108°

### Example 3

**Four interior angles of an irregular pentagon measure 100°, 120°, 90°, and 110°. Find the fifth angle.**

The five interior angles add to 540° no matter the shape.

100° + 120° + 90° + 110° = 420°

Fifth angle = 540° − 420°

**Final answer:** 120°

### Example 4

**A regular pentagon has its sides extended. Find each exterior angle.**

The interior and exterior angle at a vertex form a straight line.

Each interior angle = 108°

Each exterior angle = 180° − 108°

**Final answer:** 72°

### Example 5

**The interior angles of a pentagon are in the ratio 2 : 3 : 3 : 4 : 6. Find the largest angle.**

Let the common part be x, so the angles are 2x, 3x, 3x, 4x, 6x.

Add the parts: 2x + 3x + 3x + 4x + 6x = 18x

The total interior angle sum is 540°.

18x = 540°

x = 30°

Largest angle = 6x = 6 × 30°

**Final answer:** 180° — and this is worth pausing on. An interior angle of exactly 180° means that vertex lies flat on a straight line, so this "pentagon" is degenerate. The ratio is solvable, but it does not produce a proper five-corner pentagon. A clean problem keeps every angle below 180°.

### Example 6

**Find each interior angle of a regular pentagon using the per-angle formula directly.**

Each interior angle = [(n − 2) × 180°] ÷ n

= [(5 − 2) × 180°] ÷ 5

= (3 × 180°) ÷ 5

= 540° ÷ 5

**Final answer:** 108°

## Why The Angle Sum Never Changes

It would be easy to assume that stretching a pentagon — pulling one corner way out — changes its angle total. It does not, and the reason is the part worth keeping.

The 540° is fixed because the _number of triangles_ a pentagon decomposes into is fixed. Five sides always split into three triangles, and three triangles always carry 180° each. Stretch a corner and one triangle gets thinner while another gets wider, but they still total three triangles' worth of angle. The shape is free to change; the count of triangles is not.

This is why surveyors, tile designers, and anyone laying out a five-sided plot can check their work: measure four corners, and the fifth is forced. There is no freedom in it. The constraint that the shape must _close_ — last side meeting first — is exactly what pins the total to 540°. The same triangle-counting logic scales up to a [hexagon](/content/math/geometry/hexagon/index.html) (720°) and every larger polygon.

## Tripping Points To Avoid

### Mistake 1: Dividing 540° by the wrong number

**Where it slips in:** finding a single interior angle of a regular pentagon.

**Don't do this:** divide 540° by 3 (the triangle count) or by 4. The triangle count tells you the _total_ (3 × 180°), not the per-angle value.

**The correct way:** divide the total by the number of _angles_, which is 5. So 540° ÷ 5 = 108°. The rusher who divides by 3 gets 180° and rarely notices it is impossible.

### Mistake 2: Using 108° for an irregular pentagon

**Where it slips in:** a pentagon that is clearly not regular, but the student plugs in 108° anyway.

**Don't do this:** assume every interior angle is 108°. That value only holds when all five sides and angles are equal.

**The correct way:** for an irregular pentagon, only the _sum_ (540°) is guaranteed. Add the known angles and subtract from 540° to find the missing one. The memorizer who locks onto "pentagon = 108°" walks straight into this; the value is a special case, not a definition.

### Mistake 3: Confusing exterior angle with the angle outside the shape

**Where it slips in:** computing exterior angles.

**Don't do this:** measure the full reflex angle on the outside of the corner.

**The correct way:** the exterior angle is between one side and the _extension_ of the adjacent side, and it pairs with the interior angle to make 180°. For a regular pentagon that is 180° − 108° = 72°.

## Conclusion

- The interior angles of **any pentagon add to 540°**, from the formula (n − 2) × 180° with n = 5.

- A pentagon splits into **three triangles**, which is why the total is 3 × 180°.

- In a **regular pentagon**, each interior angle is **108°** and each exterior angle is **72°**.

- For an **irregular pentagon**, only the 540° total is fixed — find a missing angle by subtracting the known ones from 540°.

- The most common error is dividing 540° by the wrong number; always divide by **5** for a single angle.

## Frequently Asked Questions

What is the sum of the interior angles of a pentagon?

540°. This holds for every pentagon, regular or irregular, because a pentagon always divides into three triangles and each triangle contributes 180°.

What is each angle in a regular pentagon?

Each interior angle is 108° and each exterior angle is 72°.

Why is the interior angle sum (n − 2) × 180°?

Because an n-sided polygon splits from one vertex into (n − 2) triangles, and each triangle's angles add to 180°. For a pentagon, that is 3 × 180° = 540°.

Do the angles of an irregular pentagon still add to 540°?

Yes. The shape can be lopsided, but the total of all five interior angles is always 540°.

What is the difference between an interior and an exterior angle of a pentagon?

The interior angle sits inside the shape where two sides meet; the exterior angle sits between a side and the extension of the next side. At each vertex they add to 180°. You can review the broader idea on the types of angles page.