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

Frequently Asked Questions

What is the parallelogram area formula?

The area of a parallelogram is A = base × height, where the height is the perpendicular distance between the two parallel bases, not the slanted side. A parallelogram with a base of 12 m and a height of 5 m has an area of 12 × 5 = 60 m². It works because cutting a triangle off one end and sliding it to the other turns it into a rectangle.

How do you find parallelogram area without height?

If you know two neighboring sides and the angle between them, use A = a × b × sin θ. For sides of 6 and 10 meeting at a 30° angle, A = 6 × 10 × 0.5 = 30 square units. Another option is to find the height from the slanted side with the Pythagorean theorem when a right triangle is shown in the diagram.

Is parallelogram area the same as a rectangle?

A parallelogram has the same area as a rectangle with the same base and the same perpendicular height. It does not match a rectangle with the same side lengths, because leaning the sides lowers the height. A parallelogram with sides 8 and 5 and a height of 4 has area 8 × 4 = 32, while an 8 × 5 rectangle has area 40.

How do you find the area of a parallelogram with diagonals?

Use A = ½ × d₁ × d₂ × sin θ, where d₁ and d₂ are the diagonals and θ is the angle between them. With diagonals of 10 and 8 crossing at 30°, A = ½ × 10 × 8 × 0.5 = 20 square units. When the diagonals cross at 90°, as in a rhombus, sin 90° = 1 and the formula becomes ½ × d₁ × d₂.

Is the area of a parallelogram and rhombus the same?

A rhombus is a parallelogram with four equal sides, so base × height works for both. A rhombus also has the shortcut A = ½ × d₁ × d₂ because its diagonals are perpendicular. A rhombus with diagonals 6 and 8 has area ½ × 6 × 8 = 24. Its sides are 5, so its height is 24 ÷ 5 = 4.8, and 5 × 4.8 gives the same 24.

How do you find parallelogram area and perimeter?

For area, multiply the base by the perpendicular height. For perimeter, add all four sides, which is 2 × (base + side) because opposite sides are equal. A parallelogram with a base of 9 cm, a slanted side of 5 cm and a height of 4 cm has an area of 9 × 4 = 36 cm² and a perimeter of 2 × (9 + 5) = 28 cm.

Related Topics

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