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

Use S = (5-2) \cdot 180° = 540°. Since all angles in a regular pentagon are equal, each one measures 540° \div 5 = 108°.

Why does the formula use (n-2) instead of just n?

Any n-sided polygon can be divided into (n-2) non-overlapping triangles by drawing diagonals from one vertex. Since each triangle has angles summing to 180°, the total interior angle sum is (n-2) \cdot 180°.

What is the difference between interior and exterior angle sums?

Interior angles sum to (n-2) \cdot 180°, which depends on n. Exterior angles always sum to 360° regardless of the polygon's shape. The two are related at each vertex (they are supplementary), but use the correct sum for your problem.

Does this formula work for all polygons?

Yes, for any convex polygon with n \geq 3 sides. The formula (n-2) \cdot 180° applies whether you have a pentagon, hexagon, or a polygon with 100 sides. For non-convex (concave) polygons, the formula may not apply in the same way.

Sum of Interior Angles

Determines the sum of all interior angles in a polygon based on the number of sides; critical for solving angle problems in geometry.

S = (n - 2)\cdot 180^\circ

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What each symbol means

What Sum of Interior Angles takes

Symbol	Meaning
$S$	The sum of all interior angles in the polygon, measured in degrees; this is always a positive value for any valid polygon.
$n$	The number of sides (or equivalently, vertices) of the polygon; must be a whole number and at least 3 for a valid polygon.

When to use it

When you have a polygon and need to find the total of its interior angles or solve for an unknown angle.

Level

Usually taught in: Geometry · Appears on: SAT

Worked examples

1. Find the sum of interior angles of a pentagon

Problem

Find the sum of the interior angles of a pentagon.

$S = (n - 2) \cdot 180°$
Use the Sum of Interior Angles formula, where $n = 5$ for a pentagon.
$S = (5 - 2) \cdot 180°$
Substitute $n = 5$ .
$S = 3 \cdot 180°$
Evaluate the subtraction: $5 - 2 = 3$ .
$S = 540°$
Multiply: $3 \cdot 180° = 540°$ .

Answer: $540°$

A pentagon has 5 sides, and using the formula we find the sum of its interior angles is 540°. This makes sense geometrically because a pentagon divides into 3 non-overlapping triangles, each with angles summing to 180°.

2. Find the number of sides given the angle sum

Problem

A polygon has interior angles that sum to 1440°. How many sides does it have?

$(n - 2) \cdot 180° = 1440°$
Set up the equation using the Sum of Interior Angles formula.
$n - 2 = \frac{1440°}{180°}$
Divide both sides by $180°$ to isolate $(n-2)$ .
$n - 2 = 8$
Evaluate the division: $1440° \div 180° = 8$ .
$n = 10$
Add 2 to both sides to solve for $n$ .

Answer: $10$

By working backwards from the sum, we determine that the polygon has 10 sides—a decagon. We can verify: $(10-2) \cdot 180° = 8 \cdot 180° = 1440°$ , confirming our answer.

3. Find a missing interior angle in a construction layout

Problem

A construction crew is laying out a hexagonal building foundation. They have measured five of the six interior angles: 125°, 130°, 120°, 135°, and 125°. What is the sixth interior angle?

$S = (6 - 2) \cdot 180°$
For a hexagon, $n = 6$ . Use the formula to find the total sum of all interior angles.
$S = 4 \cdot 180°$
Evaluate the subtraction: $6 - 2 = 4$ .
$S = 720°$
Multiply: $4 \cdot 180° = 720°$ .
$125° + 130° + 120° + 135° + 125° = 635°$
Add the five known angles: the sum is $635°$ .
$x = 720° - 635°$
The unknown angle equals the total minus the sum of known angles.
$x = 85°$
Subtract: $720° - 635° = 85°$ .

Answer: $85°$

First we calculated the total using $S = (6-2) \cdot 180° = 720°$ . Then we subtracted all five known angles from this total to find the missing angle. This method works for any polygon as long as all but one angle is known.

Common mistakes

Where Sum of Interior Angles usually goes wrong

The mistake	Why it is wrong	The fix
$S = n \cdot 180°$	This counts too many triangles; a polygon with $n$ sides divides into exactly $n-2$ triangles, not $n$ .	Use $S = (n-2) \cdot 180°$ . For a pentagon, $S = (5-2) \cdot 180° = 540°$ , not $5 \cdot 180° = 900°$ .
For a regular pentagon, each interior angle is $\frac{180°}{5} = 36°$	This divides by $n$ directly instead of first computing the total $(n-2) \cdot 180°$ and then dividing by $n$ .	The correct angle is $\frac{(5-2) \cdot 180°}{5} = \frac{540°}{5} = 108°$ .
The interior angles sum to 720° and the exterior angles sum to 360°, so together they equal 1080°	Interior and exterior angles are separate totals and should never be added together; they measure different properties.	For a hexagon, use either $S = (6-2) \cdot 180° = 720°$ for interior angles, or remember exterior angles always sum to 360° for any polygon. Keep them separate and use each for different problems.

Tips and when to use something else

When one interior angle is unknown, use $S = (n-2) \cdot 180°$ to find the total, then subtract all known angles to find the missing one.
For regular polygons, all interior angles are equal, so each one measures $\frac{(n-2) \cdot 180°}{n}$ .
Do not confuse this with exterior angles, which always sum to 360° regardless of $n$ ; the two sums are unrelated and should never be mixed.
If you need the angle-sum for a triangle specifically, remember $(3-2) \cdot 180° = 180°$ —this formula confirms the familiar triangle angle-sum theorem.

Related formulas

Area of a Regular PolygonFind the area of any regular polygon by multiplying half the apothem by the perimeter—the quickest route when you know those two measurements.Area of a TriangleCalculates the area enclosed by a triangle given its base and perpendicular height; essential for geometry, SAT, and ACT math problems involving triangles.Special Right TrianglesSpecial Right Triangles give you the exact side ratios for 30-60-90 and 45-45-90 triangles, so you can find missing sides without the Pythagorean Theorem.

Frequently asked questions

What is the sum of interior angles of a regular pentagon?: Use $S = (5-2) \cdot 180° = 540°$ . Since all angles in a regular pentagon are equal, each one measures $540° \div 5 = 108°$ .
Why does the formula use $(n-2)$ instead of just $n$ ?: Any $n$ -sided polygon can be divided into $(n-2)$ non-overlapping triangles by drawing diagonals from one vertex. Since each triangle has angles summing to 180°, the total interior angle sum is $(n-2) \cdot 180°$ .
What is the difference between interior and exterior angle sums?: Interior angles sum to $(n-2) \cdot 180°$ , which depends on $n$ . Exterior angles always sum to 360° regardless of the polygon's shape. The two are related at each vertex (they are supplementary), but use the correct sum for your problem.
Does this formula work for all polygons?: Yes, for any convex polygon with $n \geq 3$ sides. The formula $(n-2) \cdot 180°$ applies whether you have a pentagon, hexagon, or a polygon with 100 sides. For non-convex (concave) polygons, the formula may not apply in the same way.

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Reviewed 2026-09-18