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Learn · Mathematics · Grade 10 · Geometry

Quadrilaterals and polygons

Angle sums, parallelogram, rectangle, rhombus, square, trapezoid and kite with their area formulas, and regular polygons.

High school Lesson 21 of 31 Interactive figure Practice

First, the rules

Cut it into triangles

Every polygon fact comes from triangles: draw the diagonals from one vertex and count.

★ Interior angle sum
Example 1
(n − 2) · 180° — n − 2 triangles from one vertex.
★ Exterior angles
Example 1
always 360° in total
Regular n-gon: each exterior 360° / n, each interior 180° − 360° / n.
Diagonals
Example 1
n(n − 3)2 — each vertex connects to n − 3 others; each diagonal counted twice.
★ The quadrilateral family
Example 2
Parallelogram: opposite sides parallel and equal, opposite angles equal, diagonals bisect each other. Rectangle: + right angles, equal diagonals. Rhombus: + four equal sides, perpendicular diagonals. Square: both. Trapezoid: one pair of parallel sides. Kite: two pairs of adjacent equal sides, perpendicular diagonals.
★ Areas
Examples 2, 3
parallelogram b · h · rhombus / kite d₁ · d₂2 · trapezoid (a + b)2 · h · regular polygon perimeter · apothem2

See it

Count the triangles

Change the number of sides. The fan from one vertex shows the n − 2 triangles behind (n − 2) · 180°; the orange exterior angle is 360° / n.

Step by step

Worked examples

1

A regular octagon

angles · diagonals

For a regular octagon find the sum of the interior angles, each interior angle, each exterior angle and the number of diagonals.

Ask yourself: Exterior angles are the quickest route to the interior angle of a regular polygon.

Show the solution
  1. Interior sum: (8 − 2) · 180° = 1080°.
  2. Each exterior angle: 360° / 8 = 45°; each interior: 180° − 45° = 135° (check: 1080 / 8 = 135 ✓).
  3. Diagonals: 8 · 5 / 2 = 20.

Answer 1080°, 135°, 45°, 20 diagonals

2

A rhombus from its diagonals

rhombus

A rhombus has diagonals 12 and 16. Find its area, its side and its perimeter.

ABCDO68

Ask yourself: The diagonals of a rhombus are perpendicular and bisect each other — what right triangle does that create?

Show the solution
  1. Area: 12 · 162 = 96.
  2. The diagonals cut the rhombus into four right triangles with legs 6 and 8: side √36 + 64 = 10 (a 6-8-10 triple).
  3. Perimeter 4 · 10 = 40.

Answer area 96, side 10, perimeter 40

3

A trapezoid with a right angle

trapezoid area

Trapezoid ABCD has AB ∥ DC, |AB| = 14, |DC| = 8, angle A = 90° and |AD| = 8. Find the area and the length of BC.

Ask yourself: Drop the perpendicular from C to AB: a rectangle and a right triangle appear.

Show the solution
  1. Height = |AD| = 8 (the right angle at A makes AD the height).
  2. Area: (14 + 8)2 · 8 = 88.
  3. From C drop a perpendicular to AB at E: |AE| = 8, so |EB| = 14 − 8 = 6 and |CE| = 8. Then |BC| = √36 + 64 = 10.

Answer area 88, |BC| = 10

Practice: polygons and quadrilaterals

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Angle sums, regular-polygon angles, diagonals, and the area of a parallelogram, rhombus, trapezoid or triangle. Your best score is kept in this browser only.

About this lesson

Quadrilaterals and polygons: the interior angle sum (n − 2) · 180°, exterior angles adding to 360°, regular polygons, the number of diagonals, and the family of quadrilaterals — parallelogram, rectangle, rhombus, square, trapezoid and kite — with their properties and area formulas. Worked examples and practice.

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