Area and perimeter: what they measure and how to find them

Updated October 20, 2026

Perimeter is the distance all the way around the outside of a flat shape, measured in units of length such as cm or ft. Area is the amount of surface the shape covers, measured in square units such as cm² or ft². For a rectangle, perimeter is 2 × length + 2 × width and area is length × width.

What area and perimeter measure

Imagine a rectangular garden. The fence that goes around the edge is its perimeter: a length you could measure with a tape. The soil inside, which you would cover with grass seed, is its area: a surface you could tile with unit squares.

That difference shows up in the units. Perimeter adds lengths, so a perimeter is in centimetres, metres, inches or feet. Area counts squares, so it is in square centimetres (cm²), square metres (m²), square feet (ft²) and so on. A 1 ft by 1 ft tile covers 1 ft² of floor.

The two measures are separate. A 1 by 5 rectangle and a 3 by 3 square both have a perimeter of 12, yet the rectangle covers 5 square units and the square covers 9. Knowing one does not tell you the other.

The formulas for common shapes

Each letter stands for a length: l is length, w is width, s is a side, b is a base, h is a height measured at a right angle to the base, and r is a radius. The perimeter of a circle has its own name, the circumference.

Rectangle: area = l × w, perimeter = 2l + 2w.
Square: area = s², perimeter = 4s.
Triangle: area = ½ × b × h, perimeter = the sum of the three sides.
Parallelogram: area = b × h, where h is the straight up height, not the slanted side.
Trapezoid: area = ½ × (b₁ + b₂) × h, where b₁ and b₂ are the two parallel sides.
Circle: area = πr², circumference = 2πr, which is also π times the diameter.
Any polygon: perimeter = the sum of all its sides.

Why the formulas work

A rectangle 4 units long and 3 units wide holds 3 rows of 4 unit squares, so its area is 4 × 3 = 12. Multiplying length by width is a fast way to count squares in rows.

Cut a triangle off one end of a parallelogram and slide it to the other end, and you get a rectangle with the same base and height. That is why a parallelogram's area is b × h. Two identical triangles fit together to make a parallelogram, so one triangle is half of it: ½ × b × h. A triangle with a base of 8 cm and a height of 5 cm has an area of ½ × 8 × 5 = 20 cm².

Two copies of a trapezoid, one turned upside down, make a parallelogram whose base is b₁ + b₂. Half of that parallelogram gives the trapezoid formula. For a circle, π is the number of times the diameter fits around the circumference, about 3.14, so the circumference is π times the diameter, or 2πr.

Worked example: a rectangle

Write each formula before you substitute, and keep the units with every number. It makes the difference between feet and square feet hard to miss.

ProblemA rectangular rug is 9 ft long and 6 ft wide. Find its perimeter and area.

$P$ $= 2 l + 2 w$
Perimeter adds all four sides: two lengths and two widths.
$P$ $= 2 (9) + 2 (6)$ $= 18 + 12$ $= 30 ft$
Substitute and add. Perimeter is a length, so the unit is feet.
$A$ $= l \times w$
Area counts unit squares in rows.
$A$ $= 9 \times 6$ $= 54 {ft}^{2}$
6 rows of 9 one foot squares: 54 square feet.

Answer $P = 30 ft, A = 54 {ft}^{2}$

Worked example: an L-shaped room

A shape made of rectangles is called a composite shape. For area, split it into rectangles or subtract a missing piece. For perimeter, walk around the outside and add every edge.

The perimeter of this L-shape equals the perimeter of the full 10 by 8 rectangle. Cutting a rectangular notch out of a corner moves two edges inward but does not change their total length. That is a useful check, but it only works for a notch at a corner.

ProblemA room is a 10 m by 8 m rectangle with a 4 m by 3 m corner cut out. Find its area and perimeter.

$A$ $= 10 \times 8$ $= 80 m^{2}$
Find the area of the full rectangle first.
$4 \times 3$ $= 12 m^{2}$
Find the area of the missing corner.
$A$ $= 80 - 12$ $= 68 m^{2}$
Subtract the missing piece.
$10 + 5 + 4 + 3 + 6 + 8$
Walk around the outside. The cut shortens one 10 m side to 6 m and one 8 m side to 5 m, and adds edges of 4 m and 3 m.
$P$ $= 36 m$
Add the six edges. This matches 2(10) + 2(8) = 36, as a corner notch should.

Answer $A = 68 m^{2}, P = 36 m$

Worked example: a circle

Read the question carefully: circles are often described by their diameter, but the area formula needs the radius, which is half the diameter. Keep π in the answer for an exact value, or use the π key for a decimal.

ProblemA circular plate has a diameter of 10 cm. Find its area and circumference.

$r$ $= \frac{10}{2}$ $= 5 cm$
The radius is half the diameter.
$A$ $= π r^{2}$ $= π {(5)}^{2}$ $= 25 π$
Square the radius first, then multiply by π.
$25 π$ $\approx 78.54 {cm}^{2}$
Use the π key and round to two decimal places.
$C$ $= 2 π r$ $= 2 π (5)$ $= 10 π$ $\approx 31.42 cm$
Circumference is a length, so the unit is plain centimetres.

Answer $A \approx 78.54 {cm}^{2}, C \approx 31.42 cm$

Common mistakes and how to avoid them

Most lost marks on area and perimeter come from a handful of slips:

Mixing up the two. Ask: am I going around the edge (perimeter) or covering the inside (area)?
Writing the wrong units. Perimeter is in cm, area is in cm². Writing the unit at every step catches this.
Using a slanted side as the height of a triangle or parallelogram. The height always meets the base at a right angle.
Forgetting the ½ in the triangle formula, or applying it to a rectangle.
Using the diameter in πr². Halve it first.
Adding only the labelled sides of a composite shape. Every edge counts, including the ones you have to work out.
Mixing units, such as a length in metres and a width in centimetres. Convert first, then calculate.

Where it fits in school and how to practise

In Common Core math, students count unit squares and relate area to multiplication in grade 3, use the rectangle formulas in grade 4, find the areas of triangles and other polygons in grade 6, and work with circles and composite figures in grade 7.

Practise with real objects: measure a table, a door or a book and find both its perimeter and area. To check an answer, estimate first (a 9 by 6 rug is a bit under 10 by 6, so a bit under 60 ft²), redo the area of a composite shape by splitting it a different way, and confirm the units match the question.

Questions

What is the difference between area and perimeter?

Perimeter is the distance around the outside of a shape, in units like cm. Area is the space inside it, in square units like cm².

Can two shapes have the same perimeter but different areas?

Yes. A 1 by 5 rectangle and a 3 by 3 square both have a perimeter of 12, but their areas are 5 and 9 square units. Among rectangles with a given perimeter, the square has the largest area.

Why is area measured in square units?

Area counts how many unit squares cover a surface. Multiplying two lengths, such as metres by metres, gives square metres.

How do you find the area of an irregular shape?

Split it into rectangles, triangles or other shapes you know, find each area, and add them. Or find the area of a larger shape around it and subtract the pieces that are not part of it.

Is circumference the same as perimeter?

Yes, circumference is the name for the perimeter of a circle. It equals 2πr, or π times the diameter.