Area of Rectangles and Parallelograms
The area of a rectangle is its length times its width, A = l \times w. The area of a parallelogram is its base times its height, A = b \times h. Area measures the space inside a flat shape. It is counted in square units, like square feet (ft²) or square meters (m²).
Before this lesson: You should know perimeter and how to multiply decimals and fractions.
What Area Means
Picture covering a floor with square tiles that are 1 foot on each side. The number of tiles you need is the area of the floor in square feet.
A rectangle 5 tiles long and 3 tiles wide holds 3 rows of 5 tiles. That is 3 \times 5 = 15 tiles. So you can multiply instead of counting.
Area Formulas for Rectangles, Squares and Parallelograms
| Shape | Formula | What the letters mean |
|---|---|---|
| Rectangle | A = l \times w | length times width |
| Square | A = s^2 | side times side |
| Parallelogram | A = b \times h | base times height |
A parallelogram is a four-sided shape whose opposite sides are parallel, like a slanted rectangle. Its height is the straight-up distance from the base to the top. The height makes a right angle with the base. It is not the slanted side.
Why the Parallelogram Formula Works
Cut a triangle off one end of the parallelogram and slide it to the other end. Now you have a rectangle. Its width is the same base, and its height is the same height.
Nothing was added or lost, so the area didn’t change. That is why a parallelogram uses A = b \times h, just like a rectangle.
Worked Examples
Example 1: Area of a rectangle
A rug is 9 feet long and 6 feet wide. Find its area.
Step 1. Write the formula: A = l \times w.
Step 2. Substitute and multiply: A = 9 \times 6 = 54.
Answer: The area is 54 ft².
Example 2: Area of a square
A square tile has sides of 7 cm. Find its area.
Step 1. Use A = s^2: A = 7^2 = 7 \times 7 = 49.
Answer: The area is 49 cm².
Example 3: Area of a parallelogram
A parallelogram has a base of 12 m, a height of 5 m and a slanted side of 6.5 m. Find its area.
Step 1. Pick the base and the height. The slanted side, 6.5 m, is not the height, so you don’t use it.
Step 2. Multiply: A = b \times h = 12 \times 5 = 60.
Answer: The area is 60 m².
Example 4: Find a missing side
A rectangle has an area of 84 in² and a length of 12 in. Find its width.
Step 1. Write the formula with what you know: 84 = 12 \times w.
Step 2. Divide both sides by 12: w = 84 \div 12 = 7.
Answer: The width is 7 in.
Example 5: Word problem with painting a wall
A wall is 12 feet wide and 8 feet tall. It has a window that is 3 feet wide and 4 feet tall. One can of paint covers 40 ft². How many cans do you need?
Step 1. Find the wall’s area: 12 \times 8 = 96 ft².
Step 2. Subtract the window: 96 - 3 \times 4 = 96 - 12 = 84 ft².
Step 3. Divide by the coverage: 84 \div 40 = 2.1 cans.
Step 4. You can’t buy part of a can, and 2 cans aren’t enough. Round up.
Answer: You need 3 cans of paint.
Common Mistakes with Area
| Mistake | Example of the error | Fix |
|---|---|---|
| Using the slanted side as the height | Parallelogram area 12 \times 6.5 = 78 m² | Use the height, which makes a right angle with the base: 12 \times 5 = 60 m². |
| Adding instead of multiplying | A 9 by 6 rug has area 9 + 6 = 15 | Adding is for perimeter. Area multiplies: 9 \times 6 = 54 ft². |
| Squaring as doubling | A square with side 7 has area 7 \times 2 = 14 | 7^2 = 7 \times 7 = 49. |
| Mixing units | A 2 ft by 9 in rectangle: 2 \times 9 = 18 | Use one unit: 2 ft = 24 in, so 24 \times 9 = 216 in². |
| Leaving off square units | Writing the area as 54 ft | Area uses square units: 54 ft². |
Square Units and Converting Them
One square foot is 12 inches by 12 inches, so 1 \text{ ft}^2 = 12 \times 12 = 144 \text{ in}^2. In the same way, 1 \text{ yd}^2 = 3 \times 3 = 9 \text{ ft}^2.
This is why you can’t just multiply by 12 to turn square feet into square inches. You multiply by 12 twice, once for each direction.
Practice Problems
- Find the area of a rectangle that is 8 m by 5 m.
- Find the area of a rectangle that is 10.5 in by 4 in.
- Find the area of a square with sides of 12 ft.
- A parallelogram has a base of 9 cm and a height of 4 cm. Find its area.
- A parallelogram has a base of 15 cm, a height of 7 cm and slanted sides of 8 cm. Find its area.
- A rectangle has an area of 72 m² and a width of 6 m. Find its length.
- A parallelogram has an area of 56 in² and a base of 8 in. Find its height.
- A rectangle is 2 feet long and 9 inches wide. Find its area in square inches.
- A square has an area of 81 cm². How long is each side?
- A floor is 14 ft by 11 ft. Tile costs $3 per square foot. How much will the tile cost?
Show answers
- 8 \times 5 = 40 m²
- 10.5 \times 4 = 42 in²
- 12^2 = 144 ft²
- 9 \times 4 = 36 cm²
- Use the height, not the slanted side: 15 \times 7 = 105 cm²
- 72 \div 6 = 12 m
- 56 \div 8 = 7 in
- 2 ft = 24 in, so 24 \times 9 = 216 in²
- 9 \times 9 = 81, so each side is 9 cm. (This is the square root of 81.)
- 14 \times 11 = 154 ft², and 154 \times 3 = 462. The tile costs $462.
Area of Rectangles and Parallelograms FAQ
What is the formula for the area of a rectangle?
The area of a rectangle is A = l \times w, length times width. The answer is in square units.
How do you find the area of a parallelogram?
Multiply the base by the height: A = b \times h. The height is the straight-up distance between the base and the opposite side, not the slanted side.
Is a square a rectangle?
Yes. A square is a rectangle with four equal sides, so its area is side times side, s^2.
What is the difference between area and perimeter?
Area is the space inside a shape, measured in square units. Perimeter is the distance around it, measured in regular length units.
Summary
- Area is the number of unit squares that cover a shape.
- Rectangle: A = l \times w. Square: A = s^2.
- Parallelogram: A = b \times h, using the height, not the slanted side.
- Use the same unit for both lengths, and give the answer in square units.
- 1 \text{ ft}^2 = 144 \text{ in}^2 and 1 \text{ yd}^2 = 9 \text{ ft}^2.
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