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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.

Area of trapezoid

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.
    Area of trapezoid
  • 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.
    Area of trapezoid
  • Scalene trapezoid: A trapezoid shape that has neither equal sides nor equal angles.
    Area of trapezoid

The different shape of trapezoid are:

Area of trapezoid

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:

Area of trapezoid
Area of trapezoid (A) = ½ (b1 + b2) * h

Or

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

Or

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

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.
Area of 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

A = (b1 + b2) * h

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:

Area of trapezoid (A) = ½ * h * (b1 + b2)

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

Area of trapezoid

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 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.

Area of 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.

Area of 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:

A = 6/2(12 + 8)
A = 6/2 (20)
A = 60 cm2

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.

Area of trapezoid

Solution:

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

Area of trapezoid

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.






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