Area and perimeter explained: the fence, the carpet and the formulas

Your third grader's homework shows a rectangle 6 cm long and 4 cm wide, asks for the area and the perimeter, and the answers come back as 10 and 24, or 24 and 24. Area and perimeter are two measurements of the same shape, and children mix them up because both start from the same side lengths. One picture answers what is perimeter and what is area: perimeter is the fence around a garden, and area is the carpet on the floor.

What is the difference between area and perimeter?

Perimeter is the distance around a shape, and area is the amount of flat space inside it. OpenStax Prealgebra puts it in two lines: perimeter is "a measure of the distance around a figure", and area is "a measure of the surface covered by a figure".

For a garden bed, the fence around it is its perimeter, in meters, and the soil inside is its area, in square meters.

The 6 by 4 rectangle from the homework shows both at once. Walk around the edge and you travel 6 + 4 + 6 + 4 = 20 units; count the squares inside and you get 4 rows of 6, which is 6 × 4 = 24 square units.

Why children mix the two up

Children mix up area and perimeter because both questions use the same two numbers, and a child who has not pictured the shape picks an operation at random. Adding 6 + 4 gives 10, which is neither answer, and multiplying gives 24, which is only the area.

Schools introduce both ideas in grade 3. Common Core standard 3.MD.C.7a asks children to "find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths". Standard 3.MD.D.8 then adds perimeter and asks children to show "rectangles with the same perimeter and different areas or with the same area and different perimeters", and 4.MD.A.3 moves on to "apply the area and perimeter formulas for rectangles in real world and mathematical problems".

England follows a slower path. The national curriculum programmes of study ask Year 4 pupils to "find the area of rectilinear shapes by counting squares", and Year 6 pupils to "recognise that shapes with the same areas can have different perimeters and vice versa". The overview of what math each grade should know shows where these topics sit next to multiplication and fractions.

How to find the perimeter and area of a rectangle, step by step

The perimeter of a rectangle is 2 × (length + width), and the area is length × width. Six steps take a child from the drawing to both answers with the right units.

1. Label every side. A rectangle's opposite sides are equal, so a 6 by 4 rectangle has sides 6, 4, 6 and 4. Write all four numbers on the drawing.
2. Check that the units match. A side of 2 m and a side of 50 cm must become 200 cm and 50 cm (or 2 m and 0.5 m) before you add or multiply.
3. Add all four sides for the perimeter. 6 + 4 + 6 + 4 = 20. The shortcut 2 × (6 + 4) = 2 × 10 = 20 gives the same answer.
4. Write the perimeter with a length unit. If the sides are in centimeters, the perimeter is 20 cm.
5. Multiply length by width for the area. 6 × 4 = 24. On grid paper, count the squares to confirm: 4 rows of 6 squares make 24.
6. Write the area with a square unit. The area is 24 square cm, also written 24 cm², because it counts squares that are 1 cm on each side.

Finding a missing side

A missing side is found by working the formula backward. If a rectangle has a perimeter of 30 cm and a length of 9 cm, half the perimeter is 30 ÷ 2 = 15 cm, which is length + width, so the width is 15 − 9 = 6 cm.

Area works the same way with division. The OpenStax example is a room with 168 square feet of flooring and a length of 14 feet: 168 = 14 × width, so the width is 168 ÷ 14 = 12 feet. Standard 4.MD.A.3 calls this "viewing the area formula as a multiplication equation with an unknown factor", and the guide on how to solve math word problems helps children decide which formula a story is asking for.

Perimeter vs area at a glance

Perimeter and area answer different questions about the same shape, so they use different units and different operations.

A square is a rectangle with four equal sides, so its formulas are shortcuts of the rectangle ones. A square with sides of 5 m has a perimeter of 4 × 5 = 20 m and an area of 5 × 5 = 25 m².

Can you find the perimeter from the area?

The area alone does not tell you the perimeter, because many rectangles share one area. An area of 24 square units fits all of these rectangles:

• 4 by 6: perimeter 2 × (4 + 6) = 20
• 3 by 8: perimeter 2 × (3 + 8) = 22
• 2 by 12: perimeter 2 × (2 + 12) = 28
• 1 by 24: perimeter 2 × (1 + 24) = 50

Add one side length and the answer becomes fixed. With an area of 24 and a width of 3, the length is 24 ÷ 3 = 8 and the perimeter is 22; with a square of area 36, the side is 6, because 6 × 6 = 36, and the perimeter is 4 × 6 = 24.

Same perimeter, different area: the string experiment

Rectangles with the same perimeter can cover very different areas, and a loop of string proves it at the kitchen table. Tie a string 24 cm long, stretch it into rectangles on grid paper, and count the squares inside each one.

A perimeter of 24 means length + width = 12, so the whole-number rectangles are:

• 1 by 11: area 11
• 2 by 10: area 20
• 3 by 9: area 27
• 4 by 8: area 32
• 5 by 7: area 35
• 6 by 6: area 36

The same 24 cm of fence encloses 11 square units in a long, thin shape and 36 in a square. The square always wins among rectangles, which is why a farmer with a fixed length of fencing gets the most grazing from a square field.

Every area in that list is a times-table fact. The guide to skip counting shows how counting rows of squares grows into multiplication, and the times tables game drills the facts.

Play free in your browserTimes TablesGrade 3+

L-shaped rooms and other composite shapes

The area of an L-shaped figure is found by splitting it into rectangles and adding their areas. Standard 3.MD.C.7d asks children to "find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts".

Take an L that is 6 units wide at the bottom and 5 units tall on the left. Split it into a bottom strip of 6 × 2 = 12 and a column on top of 2 × 3 = 6, so the area is 12 + 6 = 18 square units.

The perimeter needs every outside edge, including the two inner ones a child often skips. Going around the L, the edges are 2, 3, 4, 2, 6 and 5, which add to 22 units; the 4 is the top of the bottom strip (6 − 2), and the 3 is the height of the column (5 − 2).

How to check progress with area and perimeter

Five short questions show whether a child understands area and perimeter or is guessing at the operation.

• The sorting question: Ask whether a fence, a rug, a picture frame and a lawn are perimeter or area jobs. Fence and frame are perimeter; rug and lawn are area.
• The basic rectangle: Give a 7 by 3 rectangle. The perimeter is 20 and the area is 21, and the child should add the units (cm and cm²).
• The square: Give a square with sides of 9 m. The perimeter is 36 m and the area is 81 m².
• The missing side: Give an area of 40 square feet and a length of 8 feet. The width is 5 feet, found with 40 ÷ 8.
• The reasoning question: Ask which rectangle with a perimeter of 20 has the biggest area. The answer is the 5 by 5 square, with 25 square units.

A five-minute kitchen-table test

A five-minute test of three mixed questions (one rectangle, one missing side, one L-shape) shows more than a page of one type. Ask the child to draw the shape and say "around" or "inside" before calculating.

Children who get the area method right but stall on the multiplication need the facts more than the geometry. The math game for 3rd graders and the math game for 4th graders cover the grade-level number work that every area problem leans on, and the guide on how to teach long multiplication helps when sides grow to two digits.

Common area and perimeter mistakes

Four mistakes cause most wrong answers on area and perimeter problems, and each has a one-line fix.

• Adding only two sides: A 6 by 4 rectangle gets a perimeter of 10 instead of 20. Label all four sides before adding, or use 2 × (length + width).
• Multiplying for perimeter: A child writes 6 × 4 = 24 for the perimeter. Ask "around or inside?" first; around means adding.
• Mixing units: A garden 2 m by 50 cm gets an area of 100. Convert first: 200 cm × 50 cm = 10,000 cm², or 2 m × 0.5 m = 1 m².
• Forgetting square units: An area written as 24 cm reads as a length. Area always counts squares, so the answer is 24 cm² or 24 square cm.

Edge case: why area uses square units

Area uses square units because the measurement counts unit squares. Standard 3.MD.C.5a says that "a square with side length 1 unit, called 'a unit square,' is said to have 'one square unit' of area", so a rectangle covered by 24 of those squares has an area of 24 square units.

The square unit also explains why doubling the sides does not double the area. A 2 by 3 rectangle covers 6 squares, and a 4 by 6 rectangle covers 24, which is 4 times as many, while its perimeter grows from 10 to 20, only 2 times as much.

Your area and perimeter checklist and next steps

• Around or inside: decide which one the question wants before you calculate.
• All four sides: label every side of a rectangle, including the two the drawing leaves blank.
• Same units: convert every side to one unit before adding or multiplying.
• Right unit on the answer: cm for perimeter, cm² or square cm for area.
• Graph paper: draw the rectangle and count squares to confirm the product.
• Tiles or crackers: build rectangles with square tiles, then try the 24 cm string loop and find the biggest area.

The multiplication worksheet for 3rd graders gives printable practice with an answer key.

Free printable worksheetMultiplication — 3rd GradersWorksheet · PDF

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and a short daily practice. Children can also challenge 1 to 4 friends or family members in real-time math battles, in any of 33 languages.