Interior Angles: Sum Formula & Examples

TL;DR
Interior angles are the angles inside a polygon, one at each vertex. The sum of all of them is (n−2)×180° for an n-sided polygon, and in a regular polygon each one equals that sum divided by n. This article covers the definition, the sum formula and where it comes from, regular versus irregular polygons, the interior angles between parallel lines, and six worked examples.

What Are Interior Angles?

An interior angle is an angle that lies inside a polygon, formed by two of its sides meeting at a vertex. A triangle has three interior angles, a quadrilateral four, a pentagon five — one at every corner. The same phrase has a second, related meaning. When two lines are crossed by a transversal, the angles in the strip between the two lines are also called interior angles. This article focuses on the polygon meaning, which is the more common search intent, and treats the parallel-lines meaning briefly near the end. Both share the idea of "inside": inside the shape, or inside the region between two lines.

The Sum of Interior Angles Formula

Here is the rule that powers almost every problem on this topic. The sum of the interior angles of any polygon with n sides is:
S=(n−2)×180°. The formula is not a fact to memorise — it is built from the triangle. Pick any vertex of an n-sided polygon and draw diagonals from it to every other non-adjacent vertex. This slices the polygon into exactly (n−2) triangles. Each triangle contributes 180°, and together their angles make up all the polygon's interior angles, so the total is (n−2)×180°.

Run the formula across the common polygons:

A reader question that comes up constantly: what is the sum of the interior angles of a hexagon? From the formula, (6−2)×180°=720°.

One Interior Angle of a Regular Polygon

A regular polygon has all sides equal and all interior angles equal; an irregular polygon does not. The total interior angle sum is the same for both — it depends only on the number of sides — but only in a regular polygon can you find a single angle by dividing.

For a regular polygon with n sides, each interior angle is:
each interior angle=(n−2)×180°/n. So a regular pentagon has each angle 540°/5=108°, and a regular hexagon has each angle 720°/6=120°. For an irregular polygon you cannot divide like this; you add up the angles you know and subtract from the sum to find a missing one.

There is also a quick link worth holding: each interior angle and its neighbouring exterior angle sit on a straight line, so interior angle + exterior angle = 180°. That gives a second route to a regular polygon's angle — find the exterior angle (360°/n) and subtract from 180°.

Interior Angles Between Two Parallel Lines

The other place "interior angles" appears: when a transversal crosses two parallel lines, the angles in the strip between the lines are interior angles. They split into two families.

• Alternate interior angles are on opposite sides of the transversal and are equal when the lines are parallel.
• Co-interior angles (same-side interior) are on the same side of the transversal and are supplementary — they add to 180°.

A reader question — how do you solve same-side interior angles? — is answered by that supplementary rule: set the two same-side angles to sum to 180° and solve. This is a sibling topic; for the full treatment see the alternate interior angles article.

Examples of Interior Angles

With the definition, the sum formula, and the regular-polygon rule in hand, here is the topic doing real work. The problems build from a direct sum up to a missing-angle solve.

Example 1 - Find the sum of the interior angles of an octagon

An octagon has n=8 sides: S=(8−2)×180°=6×180°=1080°.

Example 2 - Find each interior angle of a regular hexagon

The correct way:
each angle=(6−2)×180°/6=720°/6=120°.

Example 3 - Four interior angles of a pentagon are 100°, 110°, 115°, and 95°. Find the fifth

The pentagon's sum is (5−2)×180°=540°. Add the four known angles: 100+110+115+95=420°. The fifth is 540°−420°=120°.

Example 4 - A regular polygon has each interior angle equal to 140°. How many sides does it have?

Each angle is (n−2)×180°/n=140°. Solve: (n−2)×180=140n, giving 40n=360 and n=9. It is a regular nonagon.

Example 5 - Find the sum of the interior angles of a polygon with 12 sides (a dodecagon)

S=(12−2)×180°=10×180°=1800°.

Example 6 - Two parallel lines are crossed by a transversal. A pair of co-interior (same-side interior) angles are (2x+20)° and (3x+10)°. Find x

Co-interior angles are supplementary, so they sum to 180°: (2x+20)+(3x+10)=180;⇒;5x+30=180;⇒;5x=150;⇒;x=30.

Why Interior Angles Matter Beyond the Classroom

The interior-angle sum is the rule that lets us build, tile, and fold the world out of flat shapes — because it tells us in advance whether shapes will fit.

• Tiling and tessellation. Tiles cover a floor with no gaps only when the interior angles meeting at a point add to exactly 360°.
• Architecture and trusses. A roof truss or a geodesic dome is a network of polygons; their interior angles must sum correctly or the structure will not close.
• Honeycombs and nature. Bees build hexagonal cells because the 120° interior angle packs the most area for the least wax.
• Game and screen design. Every polygon mesh in a 3D model relies on interior-angle sums to stay a closed surface as it is bent and rendered.

For a Grade 8 student, the interior-angle sum is the first formula that generalises — one rule covering every polygon from a triangle to a thousand-gon — which is the moment geometry starts to feel powerful rather than piecemeal.

Where Students Trip Up on Interior Angles

Mistake 1: Dividing the sum by the wrong count

Where it slips in: Finding one angle of a regular polygon, the student divides the sum by the number of triangles instead of the number of sides.
The correct way: Each angle of a regular hexagon is 720°/6=120°, not 720°/4.

Mistake 2: Using the regular-polygon formula on an irregular polygon

The correct way: Only regular polygons have equal interior angles. For an irregular polygon, the sum is still (n−2)×180°, but you find a missing angle by subtracting the known ones from the sum.

Mistake 3: Confusing interior with exterior angles

The correct way: Interior + exterior = 180° at each vertex. The exterior angle of a regular polygon is 360°/n; the interior angle is 180° minus that. Check which one the problem wants.

Key Takeaways

• Interior angles are the angles inside a polygon, one per vertex; their sum is (n−2)×180°.
• A regular polygon's single interior angle is the sum divided by n; an irregular polygon's sum is the same but its angles differ.
• Interior + exterior angle = 180° at every vertex.
• Between two parallel lines, alternate interior angles are equal and co-interior angles are supplementary.