Perimeter

Perimeter is the total distance around the outside of a flat shape. To find the perimeter of a polygon, add up the lengths of all its sides. For a rectangle that is 2 × (length + width), and for a square it is 4 × the side. The perimeter of a circle has its own name, the circumference. Perimeter is a length, so it is measured in units like inches, feet or centimeters.

What is perimeter?

Imagine walking all the way around the edge of a field, or running a piece of string around a shape and then measuring the string. That distance is the perimeter.

Perimeter answers questions like how much fence a yard needs, how much trim goes around a picture frame, or how far one lap of a track is. In the US, students start solving perimeter problems in grade 3 and use the rectangle formula in grade 4, under the Common Core standards.

Perimeter formulas

Perimeter of a rectangle

A rectangle has two lengths and two widths, so its perimeter is twice the length plus twice the width.

Example 1: a rectangle

A rectangle is 12 ft long and 7 ft wide. Find its perimeter.

The perimeter is 38 ft.

Example 2: a missing side

A rectangle has a perimeter of 50 m and a length of 16 m. How wide is it?

The width is 9 m. Check: 16 + 9 + 16 + 9 = 50.

Perimeter of a square

All four sides of a square are equal, so P = 4s. A square with sides of 9 cm has a perimeter of 4 × 9 = 36 cm. Working backward, a square with a perimeter of 52 in has sides of 52 ÷ 4 = 13 in.

Perimeter of a triangle

Add the three sides. A triangle with sides of 8 cm, 11 cm and 13 cm has a perimeter of 8 + 11 + 13 = 32 cm. For a right triangle with one side missing, find it first with the Pythagorean theorem.

Example 3: a triangle on a grid

A triangle has corners at (1, 1), (7, 1) and (7, 9). Find its perimeter.

Counting squares on the grid, the bottom side is 7 − 1 = 6 and the upright side is 9 − 1 = 8. The slanted side is the hypotenuse: √(6² + 8²) = √100 = 10. So the perimeter is 6 + 8 + 10 = 24 units.

Perimeter of a polygon

A regular polygon has all sides equal, so multiply the side by the number of sides: P = ns. A regular hexagon with sides of 6 in has a perimeter of 6 × 6 = 36 in. For an irregular polygon, add the sides one at a time.

The perimeter of a trapezoid works the same way, with the extra step of finding slanted sides from the height: see the trapezoid guide for worked examples.

Perimeter of a circle: circumference

A circle has no sides, so its perimeter is called the circumference:

A circle with a radius of 4 cm has a circumference of 2 × π × 4 = 8π ≈ 25.13 cm. Grade 7 students learn this formula under the Common Core standards.

Example 4: a semicircle

Find the perimeter of a semicircle with a radius of 5 cm. Half the circle's edge is πr = 5π, and the straight edge is the diameter, 10. So P = 5π + 10 ≈ 25.71 cm. Forgetting the straight side is the most common mistake with semicircles.

Perimeter of compound shapes

A compound shape is made of simpler shapes joined together, like an L-shaped room. Its perimeter is only the outside edge. Work out any missing sides from the ones you know, then add them all.

Example 5: an L shape

1. The right side x: the full height is 8 and the notch takes 3, so x = 8 − 3 = 5.
2. The top side y: the full width is 10 and the notch takes 4, so y = 10 − 4 = 6.
3. Add all six sides: 10 + 5 + 4 + 3 + 6 + 8 = 36.

Notice that 36 is also the perimeter of the full 10 by 8 rectangle, 2(10 + 8) = 36. Cutting a square corner out moves two sides inward but doesn't change their total length. The area goes down, but the perimeter stays the same.

Try one yourself.

Perimeter vs area

Shapes with the same perimeter can have different areas. With 20 ft of fence, you could make a 1 by 9 rectangle (area 9 ft²), a 4 by 6 rectangle (24 ft²) or a 5 by 5 square (25 ft²). For a fixed perimeter, the closer a rectangle is to a square, the more area it holds.

Perimeter word problems

Example 6: fencing a garden

A rectangular garden is 15 ft by 9 ft. Fencing costs $12 per foot. How much does it cost to fence the whole garden?

Perimeter: 2(15 + 9) = 48 ft. Cost: 48 ×12 = <strong>576.

Common mistakes

• Adding only the labeled sides. A rectangle labeled 12 and 7 has four sides, not two: the perimeter is 38, not 19.
• Mixing up perimeter and area. Perimeter adds lengths. Area multiplies them.
• Using square units. Perimeter is in ft or cm, never ft² or cm².
• Including inside lines in a compound shape. Only the outside edge counts.
• Mixing units. Change everything to the same unit first: 2 ft and 6 in is 2.5 ft, or 30 in.
• Forgetting the straight side of a semicircle or other part-circle shape.

Practice with a short timed paper

Perimeter, area and circumference come up in every middle school math course and on high school tests. The daily Pre-Algebra & Algebra 1 and Geometry, Algebra 2 & SAT Math papers are short timed practice tests of 8 questions, with a new one every day.

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