Understanding and Calculating Perimeter of Polygons - Third Grade Mathematics

Whenever you build a fence around a backyard, stitch a border around a quilt, or walk all the way around the edge of a soccer field, you are measuring perimeter. Perimeter is the total distance around the outside boundary of any two-dimensional closed figure. In this chapter, you will discover the meaning of perimeter, learn the universal addition strategy for finding perimeter on any polygon, and uncover efficient shortcut formulas for rectangles and squares.

What Is Perimeter?

The word "perimeter" comes from Greek words meaning "to measure around." Think of perimeter as walking an ant along the fence of a yard:

       Start here!
         *----------------------------+
         |                            |
         |                            |
         |         Backyard           |
         |                            |
         +----------------------------+
         Ant walks: Top + Right + Bottom + Left = Total Distance!

Perimeter is a one-dimensional, linear measurement. That means it is measured in regular linear units such as inches (in), feet (ft), centimeters (cm), or meters (m).

The Universal Rule: Add All the Outside Sides

To find the perimeter of ANY polygon—whether it has 3 sides, 4 sides, or 10 sides—simply add the lengths of all its exterior sides together!

Formula: Perimeter = Side 1 + Side 2 + Side 3 + ... + Side n

Example 1: Perimeter of a Triangle

               /              /        5 cm   /    \   5 cm
            /                 +--------+
              6 cm

Calculation: Perimeter = 5 cm + 5 cm + 6 cm = 16 centimeters.

Example 2: Perimeter of an Irregular Polygon

              3 m
           +-------+
       4 m |       | 2 m
           |       +-----+ 2 m
           |             |
           +-------------+
                 5 m

Count the sides: There are 6 sides in this polygon. Perimeter = 3 m + 2 m + 2 m + 5 m + 4 m = 16 meters.

Shortcut Formulas for Rectangles and Squares

Because rectangles and squares have special equal side properties, you can use handy shortcuts to find their perimeter more quickly than adding four separate numbers.

Perimeter of a Rectangle

In a rectangle, opposite sides are equal in length.
- Top and bottom sides are the Length (L).

- Left and right sides are the Width (W).


                   Length (L)
         +----------------------------+
         |                            |
Width (W)|                            | Width (W)
         |                            |
         +----------------------------+
                   Length (L)

Three different ways to calculate rectangle perimeter:
- Method 1 (Addition): L + W + L + W

- Method 2 (Doubles): (2 x L) + (2 x W)

- Method 3 (Half-Perimeter Doubled): 2 x (L + W)


Let us find the perimeter of a rectangle with Length = 8 inches and Width = 3 inches:
- Method 1: 8 + 3 + 8 + 3 = 22 inches.

- Method 2: (2 x 8) + (2 x 3) = 16 + 6 = 22 inches.

- Method 3: 2 x (8 + 3) = 2 x 11 = 22 inches!
All three methods yield the exact same perimeter: 22 inches.

Perimeter of a Square

In a square, all 4 sides are identical in length (s). Instead of adding s + s + s + s, you can simply multiply by 4!

Formula: Perimeter of a Square = 4 x Side

Example: A square garden has a side length of 7 feet. Perimeter = 4 x 7 ft = 28 feet!

Chapter Practice Exercises

  1. Find the perimeter of each shape: a. A triangle with side lengths 6 cm, 8 cm, and 10 cm b. A rectangle with length 9 inches and width 4 inches c. A square with side length 5 meters d. A pentagon with sides measuring 4 ft, 5 ft, 6 ft, 4 ft, and 5 ft
  2. An equilateral triangle has 3 sides of equal length. If one side is 8 cm long, what is the perimeter?
  3. A farmer wants to put a wooden fence around a rectangular goat pen that is 12 yards long and 7 yards wide. How many yards of fencing does the farmer need?
  4. A square picture frame has sides that are 10 inches long. What is the perimeter of the frame?
  5. A student measures a rectangle with sides 5 cm and 3 cm and says the perimeter is 8 cm. What mistake did the student make, and what is the true perimeter?

Solutions and Step-by-Step Answers

  1. Perimeters: a. Triangle: 6 + 8 + 10 = 24 cm. b. Rectangle: 9 + 4 + 9 + 4 = 26 inches (or 2 x (9 + 4) = 2 x 13 = 26 in). c. Square: 4 x 5 m = 20 meters. d. Pentagon: 4 + 5 + 6 + 4 + 5 = 24 feet.
  2. Equilateral triangle: 3 x 8 cm = 24 cm.
  3. Goat pen perimeter: Length = 12 yards, Width = 7 yards. Perimeter = (2 x 12) + (2 x 7) = 24 + 14 = 38 yards of fencing.
  4. Square frame: 4 x 10 inches = 40 inches.
  5. The student only added two of the sides (5 + 3 = 8), which is only half of the perimeter! A rectangle has 4 sides total. The true perimeter is 5 + 3 + 5 + 3 = 16 cm.