3rd Grade · 4th Grade · 3.MD.E.8 · 4.MD.A.3

Area and perimeter in 3rd and 4th grade

Area and perimeter are two different ways to measure a flat shape. Perimeter is the distance all the way around the edge, and area is the space inside. In 3rd grade, your child finds the perimeter of polygons and learns that it is not the same as area. In 4th grade, your child uses formulas for both to solve real problems.

What the standards say

The official Common Core wording, quoted exactly. The plain version follows.

3.MD.E.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

What it means, in plain words

3rd grade: perimeter, and how it differs from area 3.MD.E.8

Perimeter is the total length of all the sides of a shape. To find the perimeter of polygons, your child adds the lengths of the sides. A polygon is a flat, closed shape with straight sides, such as a triangle or a rectangle. Perimeter is a length, so it is measured in inches, feet or centimeters, not in square units.

Area and perimeter in 3rd grade also run backward. If your child knows the perimeter and every side but one, subtracting the known sides from the perimeter gives the missing side.

The big idea of the year is that area and perimeter can change separately. Rectangles 1 by 5, 2 by 4 and 3 by 3 all have a perimeter of 12 units. Their areas are 5, 8 and 9 square units. So two shapes can have the same perimeter and different areas, or the same area and different perimeters.

Your child should be able to:

Find the perimeter of a polygon by adding its side lengths
Find a missing side when the perimeter is known
Draw rectangles with the same perimeter and different areas
Tell whether a question asks for area or for perimeter

Example

A triangle has a perimeter of 30 inches. Two of its sides are 8 inches and 12 inches long. How long is the third side?

The two known sides add up to $8 + 12 = 20$ inches.
$30 - 20 = 10$
The third side is 10 inches long.

Example

Compare a rectangle that is 2 feet by 6 feet with a square that is 4 feet on each side. Which has the greater perimeter, and which has the greater area?

Perimeter of the rectangle: $2 + 6 + 2 + 6 = 16$ feet.
Perimeter of the square: $4 + 4 + 4 + 4 = 16$ feet.
Area of the rectangle: $2 \times 6 = 12$ square feet. Area of the square: $4 \times 4 = 16$ square feet.
The perimeters are the same, but the square has the greater area.

4th grade: using the formulas 4.MD.A.3

Area and perimeter in 4th grade use formulas for rectangles. A formula is a rule written with letters. The area is $A = l \times w$ , length times width. The perimeter is $P = 2 \times l + 2 \times w$ , because a rectangle has two lengths and two widths. Some classes write $P = 2 \times (l + w)$ , which gives the same answer.

The harder problems run a formula backward. If your child knows the area and one side, the other side is a missing factor: the number that multiplies with the known side to give the area. That turns the question into a division.

Word problems about area and perimeter of rectangles come from real life. A fence goes around a yard, so it is perimeter. Carpet covers a floor, so it is area. Your child decides which one the question asks for before picking a formula.

Your child should be able to:

Use $A = l \times w$ to find the area of a rectangle
Use $P = 2 \times l + 2 \times w$ to find the perimeter of a rectangle
Find a missing side from the area or the perimeter
Decide whether a real problem asks for area or perimeter

Example

A rectangular room has a floor area of 48 square feet. The room is 8 feet long. How wide is the room, and what is its perimeter?

Area is length times width, so $8 \times w = 48$ .
$48 \div 8 = 6$ , so the room is 6 feet wide.
Perimeter: $2 \times 8 + 2 \times 6 = 16 + 12 = 28$ .
The room is 6 feet wide, and its perimeter is 28 feet.

Where children usually get stuck

Asked how much ribbon goes around the edge of a photo 5 inches by 7 inches, your child multiplies and says 35.
Both measures are taught in the same unit, on the same drawings, so the two rules get swapped. The ribbon follows the edge, so the answer is the perimeter: 24 inches.
What helps. Have your child trace the edge with a finger for perimeter and rub the inside with a palm for area. Then ask which one the ribbon does.
Your child says two rectangles with the same perimeter must have the same area.
When two shapes match on one measure, it feels as if they must match on a related one too. Many adults use the same shortcut. For area and perimeter, it is wrong.
What helps. Tie a loop of string and stretch it into a long, thin rectangle, then into a square. The string, the perimeter, stays the same while the space inside grows.
A rectangle has an area of 42 square feet and a length of 7 feet. To find the width, your child multiplies 42 by 7.
The formula says multiply, so the child multiplies the numbers the problem gives. Here the width is the unknown factor, and finding it takes division.
What helps. Write the formula with a blank: $7 \times ? = 42$ . Ask what number fills the blank, and check it by multiplying.

Try four practice questions

Real 3rd grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Devontae wants to find the perimeter of the rectangle below.
Here is Devontae's work:
$\displaystyle 8 + 3 = 11$
Select the statement that best describes Devontae's work.
A
Devontae's $11$ comes from doubling the length but using the width only once.
B
Devontae's $11$ comes from multiplying the length by the width.
C
Devontae's $11$ comes from adding the length and the width only once.
D
Devontae's $11$ is the correct perimeter of the rectangle.
Show the answer and how to solve it
Devontae's $11$ comes from adding the length and the width only once. The four sides add to $8 + 3 + 8 + 3 = 22$ .
1. 1Look at the rectangle. Its length is $8$ and its width is $3$ .
2. 2Perimeter is the distance around the outside. Opposite sides of a rectangle are equal, so two sides are $8$ and two are $3$ . Add all four sides.
3. 3Devontae's answer is $11$ . The perimeter is $22$ , so this answer is too small.
4. 4Devontae added one length and one width. The other length and width are missing.
5. 5Select the statement: Devontae's $11$ comes from adding the length and the width only once.
Common mistake. Do not assume the work has a mistake. Count all four sides, then compare the perimeter with the answer shown.
The idea underneath. Perimeter is the distance around the outside. A rectangle has two equal lengths and two equal widths. Add all four sides to check the answer.
2On level
A rectangle has the side length and area shown below. What is the perimeter, in feet?
$\displaystyle \text{Area} = 60 \text{ square feet}$
A
17 feet
B
60 feet
C
34 feet
D
29 feet
Show the answer and how to solve it
The perimeter is $34$ feet. First find the missing side ( $60 \div 5 = 12$ ), then add all four sides: $5 + 12 + 5 + 12 = 34$ .
1. 1First, divide the area by the given side to find the missing side: $60 \div 5 = 12$ . The missing side is $12$ feet.
2. 2Next, find the perimeter by adding all four sides: $5 + 12 + 5 + 12 = 34$ . The perimeter is $34$ feet.
3. 3The answer is $34$ feet.
Common mistake. A common mistake is using the area ( $60$ ) as the perimeter, or finding the missing side correctly but then forgetting to add ALL FOUR sides — only adding two sides once gives $17$ , and doubling only the longer side gives $29$ .
The idea underneath. Area and perimeter answer different questions about a rectangle. Area uses multiplication (length times width); perimeter uses addition (all four sides). To find perimeter from area, first divide to get the missing side, then add all four sides.
3On level
What is the perimeter, in units, of the shape on the grid below?
units
Show the answer and how to solve it
Answer. 16
The perimeter is $16$ units. The rectangle is $5$ units wide and $3$ units tall, so the four sides add to $5 + 3 + 5 + 3 = 16$ units.
1. 1Look at the rectangle on the grid. The perimeter is the total length of unit segments all the way around the outside.
2. 2Find the length of each side by counting unit segments. The top and bottom sides are each 5 units long. The left and right sides are each 3 units long.
3. 3Add the four side lengths together:
Common mistake. A common mistake is counting all the unit squares inside the shape (that gives the area, 15 square units) instead of counting only the unit segments around the outside.
The idea underneath. Perimeter is the distance around a shape — count only the unit segments on the boundary, not the squares inside.
4Stretch
Carlos is putting a rope border around a rectangular swimming pool. The pool is 25 feet long and 18 feet wide. How many feet of rope does Carlos need to go all the way around it?
A
43 feet
B
68 feet
C
82 feet
D
86 feet
Show the answer and how to solve it
Carlos needs 86 feet of rope. The distance around the pool includes both lengths and both widths: $2 \times 25 + 2 \times 18 = 86$ .
1. 1Look at the pool's dimensions. The swimming pool is 25 feet long and 18 feet wide. To find the rope needed, we must find the distance all the way around it.
2. 2A rectangle has 4 sides — 2 long sides (lengths) and 2 short sides (widths). Add all four sides to find the distance around.
3. 3Group the lengths and widths together and add:
4. 4Carlos needs 86 feet of rope to go all the way around the swimming pool.
Common mistake. A common mistake is adding only one length and one width: $25 + 18 = 43$ . The distance around a rectangle includes BOTH pairs of opposite sides — both lengths AND both widths.
The idea underneath. The distance around a rectangle (its perimeter) is the sum of all 4 sides: 2 lengths plus 2 widths. The formula $P = 2 \times (l + w)$ captures this — add the length and width once, then double because there are two pairs of opposite sides.