Interior angles explained (with intuition)

Interior angles are the angles inside a polygon where two sides meet. If you walk around the edges of a polygon, every corner you turn through on the inside is an interior angle. The beautiful part of polygon geometry is that the total sum of all interior angles depends only on the number of sides, not on the polygon’s exact shape (as long as it’s a simple polygon that doesn’t self-intersect).

The key result is the classic formula: S = (n − 2) × 180°, where S is the sum of interior angles and n is the number of sides. Why does “n − 2” show up? Because you can split (triangulate) a polygon into triangles. Every triangle has interior angles that add up to 180°. If you can divide a polygon into a certain number of triangles, then the polygon’s total interior angle sum is just that number of triangles times 180°.

Here’s the intuition in slow motion:

• Pick one vertex of the polygon and draw diagonals from that vertex to every other non-adjacent vertex. You’ll break the polygon into triangles without overlap.
• A polygon with n sides will split into n − 2 triangles using that method. (Example: a pentagon splits into 3 triangles, a hexagon into 4 triangles, etc.)
• Each triangle contributes 180° of interior angles, so the polygon contributes (n − 2) × 180° in total.

Once you know the sum S, many other problems become easy: if the polygon is regular (all sides equal and all angles equal), then every interior angle is the same. So each interior angle is just the total divided by the number of angles: A = S / n = ((n − 2) × 180°) / n.

And if the polygon is not regular but you know all interior angles except one, you can solve the missing one by subtraction. You add up the angles you know, subtract from the total sum S, and the remainder is the missing angle. That’s exactly what the “Missing interior angle” mode does.

Interior angles are inside the polygon. Exterior angles are outside, created when you extend a side. People mix them up a lot because both are “angles at the vertex.” A fast way to check: if you’re looking at a normal polygon drawing (like a stop sign), the interior angles are the ones inside the shape. If you go outside the shape and measure the “turn” as you walk along the boundary, that’s exterior. This calculator is strictly for interior angles.

Worked examples you can copy

A pentagon has n = 5 sides. Use the sum formula: S = (n − 2) × 180° = (5 − 2) × 180° = 3 × 180° = 540°. So the interior angles of any pentagon (regular or irregular) always add up to 540°.

A hexagon has n = 6. First find the sum: S = (6 − 2) × 180° = 4 × 180° = 720°. If it’s regular, each angle is equal, so: A = S / n = 720° / 6 = 120°.

A quadrilateral has n = 4, so the total interior sum is: S = (4 − 2) × 180° = 360°. Suppose three angles are 95°, 110°, and 80°. Their sum is 285°. The missing angle is 360° − 285° = 75°.

With n = 5, the sum is 540°. If you know 110°, 120°, 95°, and 100°, their total is 425°. The missing angle is 540° − 425° = 115°.

If the polygon is convex (no “caved-in” corners), every interior angle should be less than 180°. If your “missing angle” comes out bigger than 180°, it might still be valid for a concave polygon, but it’s a signal to double-check the diagram.

What this calculator does step-by-step

Even though this page feels “instant,” the logic is straightforward and matches what teachers expect:

• Step 1 — Validate n: We require n ≥ 3 because a polygon needs at least three sides.
• Step 2 — Compute the interior sum: We compute S = (n − 2) × 180°. This is the foundation for every mode.
• Step 3 — Apply the selected mode:
  • Sum mode: we simply output S.
  • Regular mode: we output A = S / n, plus the sum for context.
  • Missing mode: we parse your comma-separated angles, require that you provide exactly n − 1 values, then compute Missing = S − (sum of known angles).
• Step 4 — Give a sanity-check note: We include a quick “convex vs concave” hint and show the formula used, so users can copy the solution into homework or notes.

The “meter” under the result is a visual cue for virality: it’s not a scientific chart—just a quick way to show that as you increase n, the sum of interior angles grows steadily. It’s the kind of thing people screenshot when sharing “fun geometry facts,” especially with regular polygons like a hexagon (120°) or decagon (144°).

Frequently Asked Questions

• What are interior angles?

  Interior angles are the angles inside a polygon at each vertex, formed by two adjacent sides. If you stand inside the shape and look at a corner, that corner angle is an interior angle.

• What does “regular polygon” mean?

  A regular polygon has all sides equal and all angles equal. Examples: equilateral triangle, square, regular pentagon, regular hexagon. In a regular polygon, each interior angle is S / n.

• Why is the sum formula (n − 2) × 180° always true?

  Because you can split any simple polygon into n − 2 triangles by drawing diagonals from one vertex. Each triangle has a 180° interior sum, so the polygon’s interior sum is (n − 2) × 180°.

• Does the formula work for concave polygons?

  Yes, the total sum formula still holds for simple concave polygons. However, individual interior angles can be greater than 180° in concave shapes, so your “missing angle” might be >180° and still be valid.

• Do I need to enter degrees or radians?

  This calculator uses degrees. Most interior-angle geometry worksheets use degrees by default. If your class uses radians, you can convert the final result (degrees × π/180).

• What if I know only some angles, not n−1 of them?

  The “missing angle” mode is designed for the classic situation where you know all angles except one. If you know fewer angles, there are infinitely many solutions unless additional information is given (for example, that the polygon is regular or that some angles are equal).

• Can I use this for triangles and quadrilaterals?

  Yes. Set n = 3 for triangles (sum = 180°) and n = 4 for quadrilaterals (sum = 360°). The same formulas apply—polygons are just the general case.

• How do I double-check my answer fast?

  For regular polygons, memorize a few anchors: triangle 60°, square 90°, pentagon 108°, hexagon 120°, octagon 135°, decagon 144°. If your result is far off, re-check n and whether you meant interior vs exterior angles.