When you take the perimeter of a shape, you measure around it. This is a good thing, since the word "perimeter" comes from two Greek words meaning, "to measure around."

## Perimeter Definition

Perimeter is the distance around a two-dimensional object. You can also measure the perimeter of three-dimensional objects like houses, stadiums, buildings and similar shapes.

Perimeter is the sum of the measure of every side of a 2D shape, or the sum of the measure of every side of a 3D shape measuring on the ground or floor.

Round shapes (ovals, circles, ellipses and such) have perimeter, but we almost always call that measure circumference. For any circle, the relationship between circumference, C, (the circle's perimeter) and its diameter, d, is a fixed ratio:

C = πd

Perimeter is always a linear measure. Use the same units given for each side, like mm, foot, and so on.

## Perimeter of a Triangle

The simplest polygon is the triangle, with three straight sides enclosing an area. To find the perimeter, P, of any triangle, you need to know the length each of the three sides, s:

P = s1 + s2 + s3

Most triangles are not designated with s1, s2 and s3; most illustrations use the letters a, b, and c opposite the angles A, B, and C:

[insert drawing scalene △ with sides labelled a, b, c opposite interior angles A, B, C]

A couple of shortcuts are available for perimeter of triangles. Isosceles triangles have two congruent sides called legs, so you can simply double one of them and add the third side:

[insert drawing isosceles △ with base = 19 m, legs = 21 m]

• P = s1 + s2 + s3
• P = 21 m + 21 m + 19 m
• P = 2(21 m) + 19 m
• P = 42 m + 19 m
• P = 61 m

Equilateral triangles have three congruent interior angles and therefore three congruent sides. So if you know one side, you know all three and the perimeter:

[insert drawing equilateral △ sides labelled 43 mm]

• P = s1 + s2 + s3
• P = 43 mm + 43 mm + 43 mm
• P = 3(43 mm)
• P = 129 mm

## Perimeter of a Square or Rhombus

A square or rhombus has four congruent sides. Knowing the measure of one side, s, gives you the measure of all sides. The formula is:

P = 4s

Here is a rhombus (a leaning square) with sides 37 cm. What is its perimeter?

[insert drawing rhombus with labelled sides 37 cm]

• P = 4s
• P = 4(37 cm)
• P = 148 cm

For Physical Education, your group presented a class project. You invented an outdoor game that uses a square field with sides of 17 yards. What is its perimeter?

[insert cartoon drawing square sports field sides = 17 yards; maybe put a few players on it with various pieces of sports equipment, everyone looking confused]

• P = 4s
• P = 4(17 yards)
• P = 68 yards

## Perimeter of a Rectangle or Parallelogram

The perimeter of a rectangle (or parallelogram) is found by doubling the sum of its width, w, and side length (not height), l:

• P = 2(w + l)

Here is a rectangle. What is its perimeter?

[insert drawing of labelled rectangle with long sides = 5.7 m and short sides = 4.3 m]

• P = 2(w +l)
• P = 2(5.7 m + 4.3 m)
• P = 2(20 m)
• P = 40 m

Here is a parallelogram. What is its perimeter?

[insert drawing of labelled parallelogram of any desired angle with short side length = 73 inches and width = 113 inches]

• P = 2(w + l)
• P = 2(113" + 73")
• P = 2(186")
• P = 372"

Unless the problem asks you, do not convert the answer to another unit (like feet and inches).

## Perimeter of an N-gon

What do you do when you have a pentagon (five sides), octagon (eight sides), enneadecagon (19 sides), or n-gon (n sides)? You follow the same steps as with easier polygons:

P = s1 + s2 + s3 + s4 + …

If you have the measurements, you can add to find perimeter. Proceed with caution:

• Be careful that every side uses the same measurement unit; and
• Be certain you have the measurement of every side; and
• Check to see if the shape is a square, rhombus, rectangle or parallelogram (so you do not need every side labeled); and
• Be careful to add each side only one time

An easy way to organize yourself is to pick a starting side, right its measurement, then proceed clockwise around the shape.

## Try It!

Here are some final perimeter problems to try.

1. What is the perimeter of a square with one side 89 feet long?
2. What is the perimeter of an equilateral triangle with one side measuring 197 mm?
3. And here is a tough one: The perimeter of a rectangle is 144 yards. The rectangle's long side is 49 yards. How long is the rectangle's short side?

Did you get answers of 356 feet, 591 mm, and a short side of 23 yards? We hope so!

### Next Lesson: Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
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