Achievable logoAchievable logo
AMC 10/12
Sign in
Sign up
Purchase
Textbook
Practice exams
Support
How it works
Resources
Exam catalog
Mountain with a flag at the peak
Textbook
Introduction
1. Algebra
2. Geometry
2.1 General
2.2 Similarity
2.3 Circles
2.4 Triangles
2.5 Cyclic quadrilaterals
2.6 Other quadrilaterals and polygons
2.7 3-D geometry
3. Number theory
4. Probability
5. Combinatorics
6. What's next?
Achievable logoAchievable logo
2.6 Other quadrilaterals and polygons
Achievable AMC 10/12
2. Geometry

Other quadrilaterals and polygons

1 min read
Font
Discuss
Share
Feedback

Other quadrilaterals and polygons

Back to basics: this unit is about the basic stuff that everyone supposedly knows:

  • Do you know the area formulas for all common quadrilaterals?
  • Can you easily work out the angles of arbitrarily large regular polygons?
  • How about counting their diagonals?

Examples

Example

What’s the measurement of the interior angle of a 12-gon?

(spoiler)

If you used the formula, then you get full credit. Good enough.

But if you also wanted style points for cleverness, then you divided the sum of the exterior angles (always 360o) by 12 to get that the exterior angles measure 30o each, and thus the interior angles measure 150o each.

Area formulas for common quadrilaterals

  • Rectangle: area = length × width
  • Parallelogram: area = base × height
  • Trapezoid: area = ½ × (base₁ + base₂) × height
  • Square: area = side²

Angles of regular polygons

  • Sum of interior angles: (n − 2) × 180°
  • Each interior angle: [(n − 2) × 180°] ÷ n
  • Each exterior angle: 360° ÷ n

Counting diagonals in polygons

  • Number of diagonals: n(n − 3) ÷ 2
    • n = number of sides

Example: Interior angle of a 12-gon

  • Each exterior angle: 360° ÷ 12 = 30°
  • Each interior angle: 180° − 30° = 150°

Sign up for free to take 10 quiz questions on this topic

All rights reserved ©2016 - 2026 Achievable, Inc.

Other quadrilaterals and polygons

Other quadrilaterals and polygons

Back to basics: this unit is about the basic stuff that everyone supposedly knows:

  • Do you know the area formulas for all common quadrilaterals?
  • Can you easily work out the angles of arbitrarily large regular polygons?
  • How about counting their diagonals?

Examples

Example

What’s the measurement of the interior angle of a 12-gon?

(spoiler)

If you used the formula, then you get full credit. Good enough.

But if you also wanted style points for cleverness, then you divided the sum of the exterior angles (always 360o) by 12 to get that the exterior angles measure 30o each, and thus the interior angles measure 150o each.

Key points

Area formulas for common quadrilaterals

  • Rectangle: area = length × width
  • Parallelogram: area = base × height
  • Trapezoid: area = ½ × (base₁ + base₂) × height
  • Square: area = side²

Angles of regular polygons

  • Sum of interior angles: (n − 2) × 180°
  • Each interior angle: [(n − 2) × 180°] ÷ n
  • Each exterior angle: 360° ÷ n

Counting diagonals in polygons

  • Number of diagonals: n(n − 3) ÷ 2
    • n = number of sides

Example: Interior angle of a 12-gon

  • Each exterior angle: 360° ÷ 12 = 30°
  • Each interior angle: 180° − 30° = 150°