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Interior Angle Sum
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05.20 • Polygons

Interior Angle Sum

Keep the polygon visible while you use the interior-angle sum rule so the formula stays tied to side count and triangulation.

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⬠ Interior Angle Sum
Interactive diagram

Interior Angle Sum Diagram

Change the number of sides and compare the growing total with the shape on the board.

Use the movable diagram to see what defines interior angle sum, how the labels relate to the figure, and what stays true as the board changes.

Definition: The interior angle sum of an n-gon is (n minus 2) times 180 degrees.
Detailed definition

Understanding Interior Angle Sum

Interior Angle Sum gives the total of all interior angles in an n-sided polygon. The interior angle sum of an n-gon is (n minus 2) times 180 degrees. The standard formula is one hundred eighty times the quantity n minus two.

A common explanation is triangulation. From one vertex of a convex n-gon, you can divide the polygon into n minus two triangles, and each triangle contributes one hundred eighty degrees to the total.

This formula works for convex and concave polygons alike as long as the polygon is simple. It is about the full interior total, not about the size of one single angle unless the polygon is regular.

Key facts

Important ideas to remember

  • The interior angle sum of an n-gon is (n minus 2) times 180 degrees.
  • The interior-angle sum of an n-gon is 180 times (n minus 2) degrees.
  • The formula gives the total of all interior angles, not the measure of one interior angle.
  • In a regular polygon, each interior angle is found by dividing that total equally among the n angles.
Where it is used

Where interior angle sum shows up

  • Use the interior-angle sum formula to find the total interior measure of a polygon from its side count.
  • Use it in regular-polygon problems to find one interior angle.
  • Use it in reverse when solving for the number of sides from a known total interior sum.
Common mistakes

What to watch out for

  • Do not confuse the interior-angle sum with the measure of one interior angle.
  • Do not forget to use n minus 2, not n itself, in the formula.
  • Do not apply regular-polygon equal-angle reasoning unless the polygon is stated to be regular.
Worked examples

Interior Angle Sum examples

Use these worked examples to see the idea in a clean diagram first, then in the kind of reasoning students usually need for classwork, homework, or test practice.

Example 1

Example 1: Computing the interior-angle sum from side count

Start with the number of sides, then use the correct polygon rule to find the total angle measure.

  • Count or read the number of sides.
  • Choose the correct total-angle rule.
  • Check that the resulting measure matches the type of polygon shown.

Result: The total is calculated from the side count instead of from rough estimation.

Example 2

Example 2: Comparing the interior-angle sum for two polygons

Use two different side counts to see which totals change and which totals stay fixed.

  • Read the first polygon's side count.
  • Read the second polygon's side count.
  • Compare how the total behaves in each case.

Result: The rule becomes easier to remember because the diagram shows what changes and what does not.

For

Why this page helps

This page helps because interior-angle-sum formulas become easier to remember when students see where they come from. A polygon can be broken into triangles, and the total grows by one hundred eighty degrees for each added triangle.

Do

What you can do here

  • Watch the total interior measure change as the number of sides increases.
  • Compare named polygons with the general n-gon formula on one board.
  • Download a clean angle-sum diagram or formula example for review.
Learning outcome

What this page helps you do

These takeaways are meant to help you recognize the idea faster, read diagrams more accurately, and use the topic with more confidence in real problems.

1

Interior Angle Sum

Use the interior-angle sum formula with fewer setup mistakes.

2

Interior Angle Sum

Understand why the formula grows from triangle partitions.

3

Interior Angle Sum

Move more confidently between named polygon examples and general n-gon reasoning.

05

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An n-gon is a polygon described by a general number of sides n.

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The exterior angles of a polygon always add to 360 degrees.

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