Math Core

Area and perimeter both describe a shape, but they measure very different things. Knowing which one you need helps you buy the right amount of carpet, fence or ribbon.

Inside or around?

Area is how much space covers the inside of a shape. You count it in square units.

Perimeter is the distance around the edge. You count it in plain units of length.

Look at this rectangle made of squares. Each square is 1 unit on each side.


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Area: there are 3 rows of 4 squares. $3 \times 4 = 12$ square units.
Perimeter: the sides are 4, 3, 4 and 3 units long. $4 + 3 + 4 + 3 = 14$ units.

Area vs. perimeter

Area covers the inside. Multiply length $\times$ width. The answer is in square units.
Perimeter goes around the edge. Add all the sides. The answer is in units.

Ask yourself: am I covering something or going around it? Carpet, paint and tiles cover a surface, so they need area. Fences, borders and frames go around, so they need perimeter.

Same perimeter, different areas

Suppose you have 12 units of fence. You can bend it into different rectangles. Every one of these has a perimeter of 12 units.

Length	Width	Perimeter	Area
5	1	$5 + 1 + 5 + 1 = 12$	$5 \times 1 = 5$
4	2	$4 + 2 + 4 + 2 = 12$	$4 \times 2 = 8$
3	3	$3 + 3 + 3 + 3 = 12$	$3 \times 3 = 9$

Same fence, but the areas are different! The long, skinny rectangle holds the least space. The square holds the most.

Three rectangles with a perimeter of 12 units: 5 by 1, 4 by 2 and 3 by 3.

Same area, different perimeters

Now start with 12 squares. You can arrange them into different rectangles. Every one has an area of 12 square units.

Length	Width	Area	Perimeter
12	1	$12 \times 1 = 12$	$12 + 1 + 12 + 1 = 26$
6	2	$6 \times 2 = 12$	$6 + 2 + 6 + 2 = 16$
4	3	$4 \times 3 = 12$	$4 + 3 + 4 + 3 = 14$

Same area, but the perimeters are different! The long, skinny rectangle has the biggest perimeter.

Worked example: Find both

A rectangle is 5 feet long and 2 feet wide. Find its area and its perimeter.

Area: $5 \times 2 = 10$ square feet.

Perimeter: $5 + 2 + 5 + 2 = 14$ feet.

Worked example: Rectangles with an area of 16

Find every rectangle with whole-number sides and an area of 16 square units. Which one has the smallest perimeter?

Look for pairs of numbers that multiply to 16: $1 \times 16$ , $2 \times 8$ and $4 \times 4$ .

Rectangle	Perimeter
16 by 1	$16 + 1 + 16 + 1 = 34$
8 by 2	$8 + 2 + 8 + 2 = 20$
4 by 4	$4 + 4 + 4 + 4 = 16$

The 4 by 4 square has the smallest perimeter, 16 units.

Worked example: The biggest pen

Jada has 16 meters of fence to make a rectangle-shaped pen for her rabbit. The sides must be whole meters. Which pen gives the rabbit the most room?

Each pen uses all 16 meters, so the length plus the width is $16 \div 2 = 8$ meters.

Pen	Area
7 by 1	$7$ square meters
6 by 2	$12$ square meters
5 by 3	$15$ square meters
4 by 4	$16$ square meters

The 4 by 4 pen gives the most room: 16 square meters.

Common mistake

Don't mix up the units. A 4 by 3 rectangle has an area of 12 square units and a perimeter of 14 units. If you are counting squares inside, say "square units." If you are measuring around, say "units."

Practice

Practice 1

A rectangle is 6 cm long and 2 cm wide. What is its perimeter, in centimeters?

Type a number.

Practice 2

The same rectangle is 6 cm long and 2 cm wide. What is its area, in square centimeters?

Type a number.

Practice 3

Mr. Lee wants to put a paper border around the edge of a bulletin board. What should he measure?

Practice 4

Two rectangles each have a perimeter of 16 units. One is 6 by 2. The other is 4 by 4. Which has the greater area?

Practice 5

A rectangle has an area of 18 square feet. One side is 3 feet long. What is its perimeter, in feet?

Type a number.

Practice 6

A rectangle is 5 units long and 3 units wide. Which rectangle has the same perimeter but a different area?

Practice 7

Rectangles with whole-number sides can have an area of 24 square units in several ways. Of all these rectangles, what is the smallest perimeter, in units?

Type a number.