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Polygon Angle Sums

A free Geometry lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.

A polygon with n sides can be split into n - 2 triangles, so its interior angle sum is (n - 2) x 180 degrees. One exterior angle at each vertex always adds to 360 degrees.

What you'll learn

Why it matters: Game artists and product designers use polygon angle sums when building regular tiles, icons, and low-poly models.

Worked example

Problem. Find the interior angle sum of a hexagon.

  1. A hexagon has 6 sides.
  2. Use (n - 2) x 180.
  3. (6 - 2) x 180 = 720.

Answer: 720 degrees

Practice problems

1. Find the interior angle sum of a quadrilateral.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the interior angle sum of a quadrilateral.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use (4 - 2) x 180.
  4. Check the result by substituting or estimating: the response should match 360 and make sense in the original problem.

Answer: 360

2. Find the interior angle sum of a pentagon.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the interior angle sum of a pentagon.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use (5 - 2) x 180.
  4. Check the result by substituting or estimating: the response should match 540 and make sense in the original problem.

Answer: 540

3. Find the interior angle sum of an octagon.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the interior angle sum of an octagon.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use (8 - 2) x 180.
  4. Check the result by substituting or estimating: the response should match 1080 and make sense in the original problem.

Answer: 1080

4. A regular hexagon has each interior angle equal to what measure?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A regular hexagon has each interior angle equal to what measure?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Hexagon sum is 720.
  4. 720 divided by 6 is 120.
  5. Check the result by substituting or estimating: the response should match 120 and make sense in the original problem.

Answer: 120

5. A regular polygon has 10 sides. What is each exterior angle?

Show solution
  1. Challenge: First identify exactly what the question is asking: A regular polygon has 10 sides. What is each exterior angle?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Exterior angles add to 360.
  4. 360 divided by 10 is 36.
  5. Check the result by substituting or estimating: the response should match 36 and make sense in the original problem.

Answer: 36

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