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). Three triangles means three lots of 180°.
$$\text{Sum of interior angles} = 3 \times 180° = 540°$$
That triangle-counting idea generalises into one formula for any polygon with n sides:
$$\text{Sum of interior angles} = (n - 2) \times 180°$$
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:
$$(5 - 2) \times 180° = 3 \times 180° = 540°$$
To get one interior angle of a regular pentagon, share the total across five equal corners:
$$\text{Each interior angle} = \frac{540°}{5} = 108°$$
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°:
$$\text{Each exterior angle} = 180° - 108° = 72°$$
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 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 (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.
A Practical Next Step
Practice these problems to solidify your understanding: redraw a pentagon, split it into triangles from one vertex, and confirm you count exactly three. Then take any irregular pentagon, hide one angle, and recover it from the 540° total. If you get stuck on which number to divide by, return to "Where the 540° comes from" above.
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