Perimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.
Walk around the edge and add up every side. The rectangle’s perimeter is 22 cm; the L-shape’s is 20 m. The shading inside is area, a different measurement.
Formal definition and notation
The perimeter of a polygon is the sum of the lengths of its sides. For a rectangle with length l and width w, P = 2l + 2w; for a square with side s, P = 4s; for a triangle with sides a, b and c, P = a + b + c. The perimeter of a circle has its own name, the circumference, C = 2π r. Perimeter is a one-dimensional measure given in linear units, while area is given in square units.
Symbol
Read it as
Meaning
P
P
the usual letter for perimeter
2l + 2w
two l plus two w
perimeter of a rectangle with length l and width w; also written 2(l + w)
4s
four s
perimeter of a square with side s
Key facts
Rectangles with the same area can have different perimeters: 1 by 12, 2 by 6 and 3 by 4 all have area 12, but their perimeters are 26, 16 and 14.
An L-shape built from right angles has the same perimeter as the rectangle around it: the notch moves two sides inward without changing their total length.
All sides must be in the same unit before you add them: 2 feet + 6 inches is 30 inches, not 8.
A regular polygon with n sides of length s has perimeter n × s: a regular hexagon with side 5 has perimeter 30.
Perimeter is what you measure for a fence, a frame or a border; area is what you measure for carpet, paint or grass.
Perimeter formulas
Perimeter formulas
rectangle P = 2l + 2w, square P = 4s, any polygon: add all sides
l, w
the length and width of a rectangle
s
the side of a square (all four sides are equal)
P
the perimeter, in the same unit as the sides
For shapes with unlabeled sides, use what you know first: opposite sides of a rectangle are equal, and in an L-shape the short pieces add up to the long side opposite them.
Try it: change the rectangle
Interactive
Drag the blue corner or select it and use the arrow keys. Watch the perimeter, 2(l + w), and the area change. Can you find two rectangles with the same perimeter but different areas?
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Perimeter examples, step by step
Example 1: Perimeter of a triangle
Warm-up
A triangle has sides of 7 cm, 9 cm and 11 cm. What is its perimeter?
Perimeter means add all the sides.
7 + 9 + 11 = 27.
The sides are in centimeters, so the perimeter is too.
Answer: The perimeter is 27 cm.
Example 2: Find a missing side
Core
A rectangular garden is 12 meters long and has a perimeter of 40 meters. How wide is it?
Use P = 2l + 2w: 40 = 2(12) + 2w.
2(12) = 24, so 40 = 24 + 2w.
Subtract 24: 2w = 16.
Divide by 2: w = 8. Check: 12 + 8 + 12 + 8 = 40.
Answer: The garden is 8 meters wide.
Example 3: From area to perimeter
Test-style
A square patio has an area of 49 square feet. What is its perimeter?
The area of a square is s2, so s2 = 49.
The side is s = √49 = 7 feet.
Perimeter: P = 4s = 4 × 7 = 28.
Trap: 49 is the area. Perimeter is in feet, not square feet.
Answer: The perimeter is 28 feet.
Common mistakes
✗ MistakePerimeter of an 8 cm by 3 cm rectangle = 8 × 3 = 24 cm.
✓ Instead8 + 3 + 8 + 3 = 22 cm. (24 square cm is the area.)
Do not multiply length by width for perimeter; that finds area.
✗ MistakeA 5 by 3 rectangle has perimeter 5 + 3 = 8.
✓ InsteadA rectangle has four sides: 5 + 3 + 5 + 3 = 16.
The two labeled sides are only half the edge. Opposite sides of a rectangle are equal, so count each one twice.
✗ MistakeAdding only the labeled sides of the L-shape: 6 + 4 + 3 = 13 m.
✓ InsteadFind the unlabeled sides first, then add all six: 6 + 2 + 3 + 2 + 3 + 4 = 20 m.
Perimeter includes every side of the shape, whether or not a number is written on it.
✗ MistakeThe perimeter of the 8 cm by 3 cm rectangle is 22 square cm.
✓ InsteadThe perimeter is 22 cm.
Perimeter is a length, so it uses plain length units. Square units are for area.
Area measures the surface inside a shape, in square units; perimeter measures the distance around it, in length units. An 8 by 3 rectangle has area 24 square units and perimeter 22 units.
Circumference is the perimeter of a circle. It has no straight sides to add, so you use C = 2π r or C = π d.
Quick check: 3 questions
Question 1 of 3
What is the perimeter of a square with side length 9 cm?
Show answer and explanation
Answer: C. A square has four equal sides: 4 × 9 = 36 cm. Choice A is the area, B adds only two sides, and D adds only three.
Question 2 of 3
A rectangle is 10 inches long and 4 inches wide. What is its perimeter?
Show answer and explanation
Answer: A.2(10 + 4) = 2 × 14 = 28 inches. Choice B is the area, C adds only one length and one width, and D doubles the correct answer.
Question 3 of 3
A rectangle has a perimeter of 30 cm and a width of 5 cm. What is its length?
Show answer and explanation
Answer: D.2(l + 5) = 30, so l + 5 = 15 and l = 10 cm. Choice A divides 30 by 5 as if it were an area, B subtracts both widths but forgets to halve, and C subtracts only one width.
Perimeter: frequently asked questions
Is perimeter the same as area?
No. Perimeter is the distance around a shape, and area is the amount of surface inside it. A 6 by 2 rectangle has perimeter 6 + 2 + 6 + 2 = 16 units and area 6 × 2 = 12 square units. Two shapes can share a perimeter and have different areas: a 4 by 4 square also has perimeter 16, but its area is 16.
What units is perimeter measured in?
In plain length units: inches, feet, yards, miles, centimeters, meters or kilometers, the same unit as the sides. If the sides are in meters, the perimeter is in meters. Never write square units for perimeter; those belong to area. Convert all sides to one unit before you add.
How do you find the perimeter of an irregular shape?
Add the length of every side. If some sides are not labeled, work them out first: in shapes made of right angles, the pieces along one edge add up to the full length of the opposite edge. Then go around the shape once, in order, so you do not skip or double-count a side.
PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7CircumferenceCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle’s perimeter, measured in units of length. A circle with diameter 10 has a circumference of about 31.4, because π ≈ 3.14.Geometry · Grade 6, Grade 7, Grade 8AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4InchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4Regular PolygonA regular polygon is a polygon whose sides are all the same length and whose angles are all the same size. The equilateral triangle, the square and the stop-sign octagon are regular. Each interior angle of a regular n-gon is (n − 2) × 180°n, and each exterior angle is 360°n.Geometry · Grade 5, Grade 8, GeometryCentimeterA centimeter is a metric unit of length equal to one hundredth of a meter, about the width of a fingernail. Its symbol is cm. There are 100 centimeters in 1 meter and 10 millimeters in 1 centimeter, and 1 inch equals 2.54 centimeters.Measurement · Grade 3, Grade 4, Grade 5MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2WidthWidth is the distance across a shape or object from side to side. In a rectangle, width is one of the two side measurements used with length, usually the shorter one, and multiplying length by width gives the area of a rectangle. Width is measured in length units such as inches, feet, centimeters or meters.Measurement · Grade 3, Grade 4, Grade 5FormulaA formula is a rule written with symbols that shows how quantities are related. It is an equation with two or more variables, such as A = lw for the area of a rectangle. Formulas help solve problems the same way each time: substitute the values you know, then compute the one you need.Problem Solving · Grade 5, Grade 6, Grade 7Surface AreaSurface area is the total area of all outside faces of a three-dimensional figure, measured in square units. To find it, unfold the solid into a flat net, find the area of each face and add them. A box with length l, width w and height h has surface area 2lw + 2lh + 2wh.Geometry · Grade 6, Grade 7, Grade 8VolumeVolume measures the amount of space inside a three-dimensional figure, in cubic units such as cubic centimeters or cubic feet. You can find it by counting the unit cubes that fill the solid. For a rectangular prism, V = l × w × h; for any prism, it equals the area of the base multiplied by the height.Geometry · Grade 5, Grade 6, Grade 7