Finding Unknown Side Lengths Given Perimeter - Third Grade Mathematics

Sometimes you are given the total perimeter of a shape, but one of the side lengths is missing. Finding an unknown side length turns you into a mathematical detective: you know the total distance around the entire boundary, and you must use addition, subtraction, or division to deduce the secret missing length. In this chapter, you will master finding missing side lengths for triangles, rectangles, squares, and general polygons.

The Strategy for General Polygons

For any polygon where only ONE side length is unknown:
1. Add together all the known side lengths to find their combined total.

2. Subtract that combined total from the given total perimeter.

3. The remaining difference is the length of the unknown side!


Formula: Missing Side = Total Perimeter - (Sum of Known Sides)

Example: Missing Side on a Quadrilateral

               4 cm
         +--------------+
    3 cm |              |  5 cm
         +--------------+
               ? cm
    Total Perimeter = 18 cm

Step 1: Add the known sides: 3 cm + 4 cm + 5 cm = 12 cm.

Step 2: Subtract from total perimeter: 18 cm - 12 cm = 6 cm.

The unknown bottom side is 6 cm! Check: 3 + 4 + 5 + 6 = 18 cm. The check works perfectly!

Finding Unknown Side Lengths in Rectangles

Finding a missing side of a rectangle involves two equal opposite sides.

Problem: A rectangle has a perimeter of 24 inches. Its length is 8 inches. What is its width?

                   8 in
         +----------------------+
       ? |                      | ?
         +----------------------+
                   8 in
         Perimeter = 24 in

Step-by-Step Detective Work:

  1. Identify the known sides: The rectangle has TWO lengths of 8 inches each. 8 in + 8 in = 16 inches for both lengths combined.
  2. Find the remaining perimeter for both widths: 24 in - 16 in = 8 inches remaining for both widths combined.
  3. Split the remaining distance equally between the two widths (division!): 8 inches ÷ 2 = 4 inches! The width of the rectangle is 4 inches.

Check: 8 + 4 + 8 + 4 = 24 inches. It matches!

Finding Side Lengths in Regular Polygons and Squares

When all sides of a polygon are equal, finding the side length requires only a single division step!

Formula for Square: Side Length = Perimeter ÷ 4

Example: Square with Known Perimeter

A square garden has a perimeter of 36 feet. What is the length of each side? Since a square has 4 sides of equal length: 36 ft ÷ 4 = 9 feet! Each side of the garden is 9 feet long.

Example: Equilateral Triangle with Known Perimeter

An equilateral triangle has a perimeter of 21 cm. What is the length of each side? Since an equilateral triangle has 3 equal sides: 21 cm ÷ 3 = 7 cm! Each side is 7 cm long.

Summary Strategy Table

+-----------------------+-----------------------------------------------------+
| Shape Type            | How to Find Unknown Side Length                     |
+-----------------------+-----------------------------------------------------+
| General Polygon       | Subtract sum of all known sides from perimeter      |
| Rectangle             | Subtract 2 known lengths, then divide remainder by 2|
| Square                | Divide total perimeter by 4                         |
| Equilateral Triangle  | Divide total perimeter by 3                         |
+-----------------------+-----------------------------------------------------+

Chapter Practice Exercises

  1. Find the unknown side length (?) for each figure: a. Triangle: sides are 7 cm, 5 cm, and ? cm. Perimeter = 18 cm. b. Quadrilateral: sides are 4 in, 6 in, 3 in, and ? in. Perimeter = 20 in. c. Pentagon: sides are 3 m, 4 m, 2 m, 5 m, and ? m. Perimeter = 18 m.
  2. A rectangle has a perimeter of 30 cm and a width of 5 cm. What is the length of the rectangle?
  3. A square has a perimeter of 28 inches. What is the length of each side of the square?
  4. An equilateral triangle has a perimeter of 27 feet. What is the length of each side?
  5. A rectangular swimming pool has a perimeter of 50 meters. The length of the pool is 15 meters. What is the width of the swimming pool? Show all your steps.

Solutions and Step-by-Step Answers

  1. Unknown sides: a. Known sides: 7 + 5 = 12 cm. Missing side = 18 - 12 = 6 cm. b. Known sides: 4 + 6 + 3 = 13 in. Missing side = 20 - 13 = 7 in. c. Known sides: 3 + 4 + 2 + 5 = 14 m. Missing side = 18 - 14 = 4 m.
  2. Rectangle length: Two widths: 5 + 5 = 10 cm. Remaining for two lengths: 30 - 10 = 20 cm. One length: 20 ÷ 2 = 10 cm.
  3. Square side: Perimeter = 28 inches. All 4 sides equal. Side = 28 ÷ 4 = 7 inches.
  4. Equilateral triangle: Perimeter = 27 feet. 3 sides equal. Side = 27 ÷ 3 = 9 feet.
  5. Swimming pool width:
  6. Step 1: Two lengths = 15 m + 15 m = 30 m.
  7. Step 2: Remaining perimeter for widths = 50 m - 30 m = 20 m.
  8. Step 3: Divide by 2: 20 m ÷ 2 = 10 meters. The width of the swimming pool is 10 meters.