Math Core

In third grade you found area by counting unit squares and perimeter by adding up the sides. That works, but counting gets slow for a big room or a whole soccer field. A formula is a short rule that gives you the answer in one step, no matter how big the rectangle is.

Area of a rectangle

Area is the number of square units that cover a shape. Picture a rectangle 6 units long and 4 units wide filled with unit squares. The squares make 4 rows, and each row has 6 squares. That is $4 \times 6 = 24$ squares.

You never need to draw the squares. Just multiply the length by the width.

Area of a rectangle

A = l \times w

$A$ is the area, $l$ is the length and $w$ is the width. Area is measured in square units, like square inches or square meters.

A rectangle 8 meters long and 5 meters wide.

For this rectangle, $A = 8 \times 5 = 40$ . The area is 40 square meters.

Perimeter of a rectangle

Perimeter is the distance around a shape. A rectangle has two lengths and two widths, so you can add both lengths and both widths.

Perimeter of a rectangle

P = 2 \times l + 2 \times w

Another way to say it: add the length and width, then double. $P = 2 \times (l + w)$ . Perimeter is measured in regular length units, like inches or meters.

For the rectangle above, $P = 2 \times 8 + 2 \times 5 = 16 + 10 = 26$ . The perimeter is 26 meters.

A square is a rectangle whose sides are all the same length $s$ . So its area is $s \times s$ and its perimeter is $4 \times s$ .

Worked example: Using both formulas

A rug is 9 feet long and 6 feet wide. Find its area and its perimeter.

Area: $A = 9 \times 6 = 54$ square feet.

Perimeter: $P = 2 \times 9 + 2 \times 6 = 18 + 12 = 30$ feet.

The rug covers 54 square feet, and the distance around it is 30 feet.

Common mistake

Area and perimeter are different things. Area multiplies ( $9 \times 6$ ) and uses square units. Perimeter adds all four sides ( $9 + 6 + 9 + 6$ ) and uses plain length units. Before you start, ask yourself: "Am I covering the inside, or going around the edge?"

Finding a missing side from the area

If you know the area and one side, think of the formula with a blank: $A = l \times \underline{\quad}$ . You need a number that multiplies by the length to make the area. That is a division problem.

Worked example: A missing width

A rectangular garden has an area of 72 square feet. It is 9 feet long. How wide is it?

$9 \times w = 72$ .

Ask: 9 times what equals 72? Since $72 \div 9 = 8$ , the width is 8 feet.

Check: $9 \times 8 = 72$ . ✓

Finding a missing side from the perimeter

If you know the perimeter and one side, take away the two sides you know. What is left is shared by the two matching sides.

Worked example: A missing length

A rectangular poster has a perimeter of 56 inches. Its width is 12 inches. How long is it?

The two widths use $2 \times 12 = 24$ inches.

The two lengths make up the rest: $56 - 24 = 32$ inches.

Each length is $32 \div 2 = 16$ inches. The poster is 16 inches long.

Check: $2 \times 16 + 2 \times 12 = 32 + 24 = 56$ . ✓

Worked example: A real-world problem

A classroom floor is 30 feet long and 20 feet wide. Carpet squares each cover 1 square foot. The teacher also wants a border strip all the way around the edge. How many carpet squares and how many feet of border are needed?

Carpet covers the inside, so find the area: $30 \times 20 = 600$ square feet. That is 600 carpet squares.

The border goes around the edge, so find the perimeter: $2 \times 30 + 2 \times 20 = 60 + 40 = 100$ feet of border.

Tip

A quick check for a missing side: put your answer back into the formula. If you get the area or perimeter you started with, you are right.

Practice

Practice 1

A rectangle is 9 cm long and 7 cm wide. What is its area, in square centimeters?

Type a number.

Practice 2

A rectangle is 11 inches long and 5 inches wide. What is its perimeter, in inches?

Type a number.

Practice 3

A square tile has sides that are 12 centimeters long. What is its perimeter, in centimeters?

Type a number.

Practice 4

Jin wants to know how much paint he needs to cover a wall. Should he find the wall's area or its perimeter?

Practice 5

A rectangle has an area of 56 square meters. Its length is 8 meters. What is its width, in meters?

Type a number.

Practice 6

A rectangular pen has a perimeter of 46 feet. Its width is 9 feet. What is its length, in feet?

Type a number.

Practice 7

A soccer field is shaped like a rectangle 50 meters long and 30 meters wide. What is its area, in square meters?

Type a number.

Practice 8

A rectangular window has an area of 24 square feet. Its width is 3 feet. What is its perimeter, in feet?

Type a number.