Perimeter of Polygons & Missing Sides: Fifth Grade Mathematics

Perimeter measures the total distance around the outside edge of a closed two-dimensional polygon, calculated by summing all exterior side lengths.

The Definition of Perimeter

The perimeter of any polygon is found by adding the lengths of all its exterior sides: Perimeter = Side 1 + Side 2 + Side 3 + ... + Side n. Perimeter is a one-dimensional linear measurement expressed in linear units (such as inches, centimeters, feet, or meters).

+--------------------------------------------------------------------------+
|                  PERIMETER FORMULAS FOR SPECIAL POLYGONS                 |
+---------------------+----------------------------+-----------------------+
| Shape               | Perimeter Formula          | Example               |
+---------------------+----------------------------+-----------------------+
| Any Polygon         | P = Sum of all sides       | Triangle: s1 + s2 + s3|
| Rectangle           | P = 2l + 2w  or  2(l + w)  | l = 8, w = 5 -> P = 26|
| Square              | P = 4s                     | s = 6 -> P = 24       |
| Regular n-gon       | P = n * s                  | Regular Hexagon: 6 * s|
+---------------------+----------------------------+-----------------------+

Visualizing Perimeter as an Unrolled String:
       Side 1 (4 cm)     Side 2 (3 cm)     Side 3 (5 cm)
    +-----------------+-----------------+-----------------------+
    <----------------------- Total String: 12 cm ---------------------->

Finding a Missing Side Length

When the total perimeter is known along with some side lengths, you can find the missing side by subtracting the known sides from the perimeter.

Finding Missing Side x of a Rectangle:
  Given: Perimeter = 38 cm, Length = 12 cm, Width = x
  Step 1: Account for two lengths: 12 + 12 = 24 cm.
  Step 2: Subtract lengths from total perimeter: 38 - 24 = 14 cm remaining.
  Step 3: Divide remaining distance between the two widths: 14 / 2 = 7 cm.
  Width x = 7 cm.

Exercises and Solutions

Exercise 1: Missing Side in an Irregular Polygon. A pentagon has a total perimeter of 48 centimeters. Four of its side lengths measure 7 cm, 11 cm, 9 cm, and 13 cm. Find the length of the fifth side.

Solution 1: Add the four known side lengths: 7 + 11 + 9 + 13 = 40 cm. Subtract this sum from the total perimeter of 48 cm: 48 - 40 = 8 cm. The fifth side length is 8 centimeters.

Exercise 2: Finding Dimensions from Perimeter. A rectangular playground has a perimeter of 96 meters. The length of the playground is 30 meters. Find the width of the playground and calculate its area.

Solution 2: Use the rectangle perimeter formula P = 2l + 2w: 96 = 2(30) + 2w. Multiply 2 * 30 = 60. Subtract 60 from 96: 96 - 60 = 36 meters. Divide by 2 to find the width: w = 36 / 2 = 18 meters. The width is 18 meters. To find the area, multiply length by width: Area = 30 * 18 = 540 square meters.