Area of a Parallelogram

How to area of a parallelogram, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

A parallelogram is a four-sided figure in which opposite sides are parallel and equal in length. Rectangles, rhombuses, and squares are all special cases of parallelograms. The formula for the area of a parallelogram is simple, base times height, and mastering it helps you solve many GED geometry problems.

What Is a Parallelogram?

A parallelogram is a quadrilateral with two pairs of parallel sides. Key properties:

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• Opposite sides are equal in length and parallel.
• Opposite angles are equal.
• Consecutive angles are supplementary (sum to 180°).
• The height h is the perpendicular distance between the two parallel bases. It is not the slant side length.

Area Formula

(color{blue}{A = b times h})

• b = length of the base (either pair of parallel sides)
• h = perpendicular height (the altitude, drawn at 90° to the base)

Why does this work? If you slice a triangle off one end of a parallelogram and slide it to the other end, you form a rectangle with the same base and height, so the areas are equal.

How to Find the Area

Step 1: Identify the base

Choose either pair of parallel sides as the base.

Step 2: Find the perpendicular height

The height is the perpendicular distance between the two parallel sides. It is not the slanted side.

Step 3: Multiply (color{blue}{text{ base } times text{ height }})

Apply the formula and include square units.

Step-by-Step Summary

1. Identify the base b (any side) and the corresponding perpendicular height h.
2. Calculate (color{blue}{A = b times h}).
3. Label the answer in square units.

Watch: Area of a Parallelogram (Video Lesson)

Math with Mr. J explains how to find the area of a parallelogram with step-by-step examples:

Worked Examples

Example 1: Find the area of a parallelogram with base 8 cm and height 5 cm.

(color{blue}{A = 8 times 5 = 40 text{ cm }^{2}})

Example 2: A parallelogram has a base of 12 ft and a perpendicular height of 6 ft. Find the area.

(color{blue}{A = 12 times 6 = 72 text{ ft }^{2}})

Example 3: A parallelogram-shaped tile has a base of 7 inches and a slant side of 5 inches. The perpendicular height is 4 inches. What is the area?

Use the perpendicular height (4 in), not the slant side.
(color{blue}{A = 7 times 4 = 28 text{ in }^{2}})

Example 4: The area of a parallelogram is 30 cm² and the base is 10 cm. What is the height?

(color{blue}{30 = 10 times h rightarrow h = 30 div 10 = 3 text{ cm }})

More Practice: Area of a Parallelogram (Video)

Khan Academy explains the formula and works through several geometry examples:

Exercises

1. Find the area of a parallelogram with base 9 m and height 4 m.
2. A parallelogram has base 15 in. And height 7 in. Find its area.
3. The area of a parallelogram is 56 cm² and the height is 7 cm. Find the base.
4. A parking space is parallelogram-shaped with a base of 8 ft and height of 18 ft. Find the area.
5. A rhombus has a base of 6 cm and a perpendicular height of 5 cm. Find its area.
6. Two parallelograms have the same height of 6 m. One has base 5 m; the other has base 10 m. What is the ratio of their areas?

Answers

1. (color{blue}{9 times 4 = 36 m^{2}})
2. (color{blue}{15 times 7 = 105 text{ in }^{2}})
3. (color{blue}{b = 56 div 7 = 8 text{ cm }})
4. (color{blue}{8 times 18 = 144 text{ ft }^{2}})
5. (color{blue}{6 times 5 = 30 text{ cm }^{2}}) (a rhombus is a parallelogram)
6. A₁ = 30; A₂ = 60; ratio = 1:2

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Frequently Asked Questions

Is the height of a parallelogram the same as its side length?

No, only in a rectangle (where the sides are perpendicular to the base). In a general parallelogram, the sides are slanted. The height is the perpendicular distance between the parallel sides, which is always less than or equal to the slant side.

Does a rectangle use the same area formula?

Yes. A rectangle is a special parallelogram where the height equals the side length. So (color{blue}{A = l times w}) for a rectangle is the same as (color{blue}{A = b times h}) for a parallelogram.

How is the parallelogram formula related to the triangle formula?

The area of a triangle is half the area of a parallelogram with the same base and height. This is because any parallelogram can be split into two congruent triangles along a diagonal.

Related Topics

• Area of a Triangle
• Area of a Trapezoid
• Polygons
• Triangles
• Finding Area of Compound Figures