Area of trapezoid

The trapezoid is a convex quadrilateral shape. It means that a closed-shape that has four sides with one pair of parallel sides.

In this section, we will learn how to find the area of a trapezoid.

Trapezoid

The trapezoid is a gematrical shape that has four-sides in which a pair of opposite sides must be parallel is called trapezoid. The parallel sides are known as base, and the non-parallel sides are known as leg. The parallel sides can be horizontal, vertical, or slanting (diagonal). It is also known as trapezium.

In the following figure, base₁ and base₂ are perpendicular to the dotted line, which represents the altitude (height). The altitude is perpendicular to the distance between two bases.

Properties of a Trapezoid

• A trapezoid is a parallelogram if both pairs of its opposite sides are parallel.
• A trapezoid is a square if both pairs of its opposite sides are parallel; all its sides are of equal length and at right angles to each other.
• A trapezoid can be a rectangle if both pairs of its opposite sides are parallel; its opposite sides are of equal length and are at right angles to each other.
• Each base must be perpendicular to the height.

Types of Trapezoid

There are three types of trapezoid:

• Right trapezoid: It contains a pair of right angles.
  
• Isosceles trapezoid: A trapezoid shape that has an equal length of non-parallel sides. In the following image, the sides AD and BC are of the same length.
  
• Scalene trapezoid: A trapezoid shape that has neither equal sides nor equal angles.
  

The different shape of trapezoid are:

To find the area of the trapezoid follows the steps below:

• Add the bases
• Multiply the sum by height
• Divide the result by 2

Area of Trapezoid Formula

The area of a trapezoid is the average width times the altitude. On implementing it in the formula:

Or

Or

Where b1 and b2 are the lengths of each base, and h represents the altitude.

Derivation of Trapezoid Formula

Let's derive the formula by using the parallelogram from two congruent trapezoids.

• Draw a trapezoid whose base length is b1 and b2, and altitude is h.
• Create a copy of it.
• Rotate the copy of the trapezoid by .
• Add a copy of the trapezoid to the original trapezoid.

When we combine the two trapezoids (as above), it forms a parallelogram shape. The newly formed shape has two pairs of opposite congruent sides.

We know that:

area of a parallelogram (A) = b * h

It means the area of a parallelogram is altitude (height) times the length of either base. From the above figure, the length of both the bases is equal to b1 + b2.

According to the formula of the parallelogram, we get

The above area is the area of two trapezoids. So, we need to divide it by 2 to get the area of one trapezoid.

Area of trapezoid (A) = (b1 + b2) * h / 2

Arranging the above formula, we get:

Another way to calculate the area of a trapezoid by dividing the trapezoid into two triangles.

Now we have two triangles ∆ABD and ∆BCD. We will calculate the area of the triangles separately. The total area of the triangles will be the area of the trapezoid.

Area of ∆

A = ½ b1 * h

Area of ∆

A = ½ b2 * h

Adding the area, we get:

A + A = ½ (b1 * h) + ½ (b2 * h)

2A = (b1 + b2) * h

A = 1/2 (b1 + b2) * h

Examples

Example 1: If the bases of a trapezoid are 30 inches and 20 inches. The height of trapezoid is 4 inches. Find the area of the trapezoid.

Solution:

Given, b1 = 30 inches, b2 = 20 inches and height = 4 inches

Area of trapezoid (A) = 1/2 (b1 + b2) * h

A = ½ (30 + 20) * 4

A = ½ (50) * 4

A = 200/2

A = 100 in²

The area of trapezoid is 100 in².

Example 2: Find the area of the given trapezoid.

Solution:

From the above figure, we will consider only parallel sides AB and CD because it makes the right angle with the parallel sides.

Given, b1=12 cm, b2=8 cm and h=6 cm

We know that,

A = h/2 (b1 + b2)

Putting the values in the above formula, we get:

The area of the trapezoid is 60 cm².

Example 3: Find the area of a trapezoid whose bases are 10 m and 7 m, respectively. The altitude of the trapezoid is 5 m. Find the area.

Solution:

First, we will divide the trapezoid into two triangles, as we have shown in the following figure.

Now find the area of triangles separately. We know that:

Area of triangle (A) = 1/2 b * h

Area of ∆ABC = ½ (7 * 5)

A = 35/2

A = 17.5 m²

Similarly, the area of ∆CDA = ½ (10 * 5)

A = 50/2

A = 25 m²

Adding the area of ∆ABC and ∆CDA we get the total area of the trapezoid.

Area of trapezoid = Area of ∆ABC + Area of ∆CDA

Area of trapezoid (A) = 17.5 + 25

A = 42.5 m²

The area of the trapezoid is 42.5 m².