DWG NO. 10.2 — Lesson 2 of 5

Area of Trapezoids, Rhombi, and Kites

Unit 10: Area of Polygons and Circles · ~15–30 min

Objective

Find the area of a trapezoid from its parallel sides and height, and the area of a rhombus or kite from its diagonals.

Trapezoids: averaging the parallel sides

A trapezoid has exactly one pair of parallel sides, called bases (b₁ and b₂), and a height measured perpendicular between them. Its area is the average of the two bases, times the height — equivalent to treating the trapezoid as a rectangle with a "typical" width halfway between the two base lengths.

Trapezoid: Area = ½(b₁ + b₂)h.

The trapezoid's height is perpendicular to both parallel bases b₁ and b₂.

Rhombi and kites: half the diagonal product

Both a rhombus and a kite have perpendicular diagonals, and that perpendicularity is what makes a single simple formula work for both shapes: multiply the diagonals and take half.

Rhombus or Kite: Area = ½d₁d₂, where d₁ and d₂ are the lengths of the two diagonals.

This works because the perpendicular diagonals split the quadrilateral into four right triangles that, rearranged, exactly fill a rectangle with side lengths d₁ and d₂ — so the quadrilateral's area is half that rectangle's area.

The perpendicular diagonals d₁ and d₂ determine the area of a rhombus or kite.

Worked Example 1 · Trapezoid area

ProblemA trapezoid has bases 10 cm and 16 cm, and height 7 cm. Find its area.

1.A = ½(b₁ + b₂)h = ½(10 + 16)(7).

2.A = ½(26)(7) = ½(182) = 91.

Area = 91 cm²

Worked Example 2 · Rhombus from diagonals

ProblemA rhombus has diagonals 12 in and 18 in. Find its area.

1.A = ½d₁d₂ = ½(12)(18).

2.A = ½(216) = 108.

Area = 108 in²

Guided practice

1.Find the area of a trapezoid with bases 8 m and 14 m and height 5 m.
Show answer
A = ½(8 + 14)(5) = ½(22)(5) = ½(110) = 55 m².
2.A kite has diagonals 9 ft and 20 ft. Find its area.
Show answer
A = ½(9)(20) = ½(180) = 90 ft².
3.A rhombus has area 150 cm² and one diagonal of 25 cm. Find the other diagonal.
Show answer
150 = ½(25)(d₂) = 12.5d₂, so d₂ = 12 cm.
4.Explain why the rhombus/kite area formula requires the diagonals to be perpendicular.
Show answer
The formula comes from splitting the quadrilateral into four right triangles that rearrange into a d₁ by d₂ rectangle; that rearrangement only works cleanly when the diagonals cross at a right angle.
5.A trapezoid's longer base is twice its shorter base. The shorter base is 9 in and the height is 6 in. Find the area.
Show answer
Longer base = 18 in. A = ½(9 + 18)(6) = ½(27)(6) = ½(162) = 81 in².

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