Area of a Parallelogram
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 a wide range of GED geometry problems.
What Is a Parallelogram?
A parallelogram is a quadrilateral with two pairs of parallel sides. Key properties:
- 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
- Identify the base b (any side) and the corresponding perpendicular height h.
- Calculate \(\color{blue}{A = b \times h}\).
- 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
- Find the area of a parallelogram with base 9 m and height 4 m.
- A parallelogram has base 15 in. and height 7 in. Find its area.
- The area of a parallelogram is 56 cm² and the height is 7 cm. Find the base.
- A parking space is parallelogram-shaped with a base of 8 ft and height of 18 ft. Find the area.
- A rhombus has a base of 6 cm and a perpendicular height of 5 cm. Find its area.
- 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
- \(\color{blue}{9 \times 4 = 36 m^{2}}\)
- \(\color{blue}{15 \times 7 = 105 \text{ in }^{2}}\)
- \(\color{blue}{b = 56 \div 7 = 8 \text{ cm }}\)
- \(\color{blue}{8 \times 18 = 144 \text{ ft }^{2}}\)
- \(\color{blue}{6 \times 5 = 30 \text{ cm }^{2}}\) (a rhombus is a parallelogram)
- A1 = 30; A2 = 60; ratio = 1:2
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.
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