Learn on PengiBig Ideas Math, Course 1Chapter 4: Areas of Polygons

Lesson 3: Areas of Trapezoids

In this Grade 6 lesson from Big Ideas Math Course 1, students learn how to derive and apply the trapezoid area formula A = ½h(b₁ + b₂) using the height and two bases. Through a hands-on grid paper activity, students cut and rearrange a trapezoid into familiar shapes to understand where the formula comes from before practicing with numerical and grid-based examples. The lesson also connects the formula to real-life applications such as estimating geographic area and population density.

Section 1

Properties and Area of Trapezoids

Property

A trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases.
The area $A$ of a trapezoid with parallel bases $b_1$ and $b_2$ and perpendicular height $h$ is given by the formula:

A = \frac{1}{2}h(b_1 + b_2)

Examples

A trapezoid has a height of 8 inches and bases of 10 inches and 15 inches. The area is $A = \frac{1}{2} \cdot 8 \cdot (10 + 15) = 4 \cdot 25 = 100$ square inches.

Find the area of a trapezoid with a height of 4 feet and bases of 7.5 feet and 11.5 feet. The area is $A = \frac{1}{2} \cdot 4 \cdot (7.5 + 11.5) = 2 \cdot 19 = 38$ square feet.

Section 2

Area of a Trapezoid

Property

The area of a trapezoid can be found by splitting it into two triangles. Let the lengths of the two parallel sides (the bases) be $a$ and $b$ , and let the perpendicular distance between them be the height, $h$ . The area is the sum of the areas of the two triangles:

\operatorname{Area} = \frac{1}{2} a h + \frac{1}{2} b h = \frac{1}{2} (a + b) h

Examples

A trapezoid has parallel bases of 6 inches and 10 inches, and a height of 5 inches. Its area is $\frac{1}{2}(6 + 10) \times 5 = 40$ square inches.
A garden plot is a trapezoid with parallel sides of 8 ft and 12 ft. The height is 7 ft. The area is $\frac{1}{2}(8 + 12) \times 7 = 70$ square feet.
A window is shaped like a trapezoid with bases of 20 cm and 30 cm and a height of 15 cm. Its area is $\frac{1}{2}(20 + 30) \times 15 = 375$ square cm.

Explanation

To find a trapezoid's area, you can average its two parallel bases and multiply by the height. This works because you can slice a trapezoid into two triangles, and this formula is just a shortcut for adding their areas together.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Properties and Area of Trapezoids

Property

A trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases.
The area $A$ of a trapezoid with parallel bases $b_1$ and $b_2$ and perpendicular height $h$ is given by the formula:

A = \frac{1}{2}h(b_1 + b_2)

Examples

A trapezoid has a height of 8 inches and bases of 10 inches and 15 inches. The area is $A = \frac{1}{2} \cdot 8 \cdot (10 + 15) = 4 \cdot 25 = 100$ square inches.

Find the area of a trapezoid with a height of 4 feet and bases of 7.5 feet and 11.5 feet. The area is $A = \frac{1}{2} \cdot 4 \cdot (7.5 + 11.5) = 2 \cdot 19 = 38$ square feet.

Section 2

Area of a Trapezoid

Property

\operatorname{Area} = \frac{1}{2} a h + \frac{1}{2} b h = \frac{1}{2} (a + b) h

Examples

A trapezoid has parallel bases of 6 inches and 10 inches, and a height of 5 inches. Its area is $\frac{1}{2}(6 + 10) \times 5 = 40$ square inches.
A garden plot is a trapezoid with parallel sides of 8 ft and 12 ft. The height is 7 ft. The area is $\frac{1}{2}(8 + 12) \times 7 = 70$ square feet.
A window is shaped like a trapezoid with bases of 20 cm and 30 cm and a height of 15 cm. Its area is $\frac{1}{2}(20 + 30) \times 15 = 375$ square cm.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Properties and Area of Trapezoids

Property

A trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases.
The area $A$ of a trapezoid with parallel bases $b_1$ and $b_2$ and perpendicular height $h$ is given by the formula:

A = \frac{1}{2}h(b_1 + b_2)

Examples

A trapezoid has a height of 8 inches and bases of 10 inches and 15 inches. The area is $A = \frac{1}{2} \cdot 8 \cdot (10 + 15) = 4 \cdot 25 = 100$ square inches.

Find the area of a trapezoid with a height of 4 feet and bases of 7.5 feet and 11.5 feet. The area is $A = \frac{1}{2} \cdot 4 \cdot (7.5 + 11.5) = 2 \cdot 19 = 38$ square feet.

Section 2

Area of a Trapezoid

Property

\operatorname{Area} = \frac{1}{2} a h + \frac{1}{2} b h = \frac{1}{2} (a + b) h

Examples

A trapezoid has parallel bases of 6 inches and 10 inches, and a height of 5 inches. Its area is $\frac{1}{2}(6 + 10) \times 5 = 40$ square inches.
A garden plot is a trapezoid with parallel sides of 8 ft and 12 ft. The height is 7 ft. The area is $\frac{1}{2}(8 + 12) \times 7 = 70$ square feet.
A window is shaped like a trapezoid with bases of 20 cm and 30 cm and a height of 15 cm. Its area is $\frac{1}{2}(20 + 30) \times 15 = 375$ square cm.