Finding Unknown Sides Using Perimeter - Complete Guide

Master how to find missing side lengths when the perimeter is known. Learn through detailed exercises, solutions, and visual examples.

Exercise Solutions 1 to 3
1 Square Unknown Side
Exercise 1
A square has a perimeter of 24 cm. What is the length of each side?
Definition:

Perimeter: The total distance around the outside of a shape

Unknown side: A side length that needs to be calculated

Square: A four-sided shape with all sides equal

Method for Finding Unknown Sides:
  1. Identify what you know (perimeter and number of sides)
  2. Set up the equation: Perimeter = Number of sides × Side length
  3. Rearrange to solve for the unknown: Side length = Perimeter ÷ Number of sides
Known Perimeter
24 cm
Number of Sides
4 sides
Unknown Side
24 ÷ 4 = 6 cm
Step 1: Identify what you know

Perimeter = 24 cm, Square has 4 equal sides

Step 2: Set up the equation

Perimeter = Number of sides × Side length

24 = 4 × Side length

Step 3: Solve for the unknown

Side length = 24 ÷ 4 = 6 cm

Step 4: Verify the answer

6 + 6 + 6 + 6 = 24 cm ✓

Each side = 6 cm
Final answer:

Each side of the square is 6 cm long

Applied rules:

• Division Rule: To find one part, divide total by number of parts

• Perimeter Formula: P = n × s, so s = P ÷ n

• Verification: Add all sides to confirm result

Tip: For a square, divide the perimeter by 4 to get side length.
2 Triangle Unknown Side
Exercise 2
An equilateral triangle has a perimeter of 21 cm. What is the length of each side?
Definition:

Equilateral triangle: A triangle with all three sides equal

Triangle: A three-sided polygon

Known Perimeter
21 cm
Number of Sides
3 sides
Unknown Side
21 ÷ 3 = 7 cm
Step 1: Identify what you know

Perimeter = 21 cm, Triangle has 3 equal sides

Step 2: Set up the equation

21 = 3 × Side length

Step 3: Solve for the unknown

Side length = 21 ÷ 3 = 7 cm

Step 4: Verify

7 + 7 + 7 = 21 cm ✓

Each side = 7 cm
Final answer:

Each side of the triangle is 7 cm long

Applied rules:

• Division Rule: Side length = Perimeter ÷ Number of sides

• Equilateral Property: All sides are equal

• Triangle Property: Has 3 sides

Tip: For an equilateral triangle, divide the perimeter by 3.
3 Hexagon Unknown Side
Exercise 3
A regular hexagon has a perimeter of 30 cm. What is the length of each side?
Definition:

Regular hexagon: A six-sided polygon with all sides equal

Hexagon: A six-sided polygon

Known Perimeter
30 cm
Number of Sides
6 sides
Unknown Side
30 ÷ 6 = 5 cm
Step 1: Identify what you know

Perimeter = 30 cm, Hexagon has 6 equal sides

Step 2: Set up the equation

30 = 6 × Side length

Step 3: Solve for the unknown

Side length = 30 ÷ 6 = 5 cm

Step 4: Verify

5 + 5 + 5 + 5 + 5 + 5 = 30 cm ✓

Each side = 5 cm
Final answer:

Each side of the hexagon is 5 cm long

Applied rules:

• Division Rule: Side length = Perimeter ÷ Number of sides

• Regular Polygon Property: All sides are equal

• Hexagon Property: Has 6 sides

Tip: For any regular polygon, divide the perimeter by the number of sides.
Key Formulas, Definitions & Properties
s = P ÷ n
Finding Unknown Side Length
Key Definitions:

Perimeter (P): The total distance around the outside of a shape

Side Length (s): The measurement of one side of the polygon

Number of Sides (n): The count of sides in the polygon

Unknown Side: A side length that needs to be calculated

Regular Polygon: A shape with all sides equal and all angles equal

General Formula
s = P ÷ n
Side = Perimeter ÷ Number of sides
Square
s = P ÷ 4
Divide by 4
Triangle
s = P ÷ 3
Divide by 3
Hexagon
s = P ÷ 6
Divide by 6
Regular Polygon Properties: All sides equal, all angles equal, symmetry
Unknown Side Calculation: Perimeter ÷ Number of sides = Side length
Tip 1: Count the number of sides carefully before dividing.
Tip 2: Always verify by multiplying the answer back to check perimeter.
Tip 3: Remember: s = P ÷ n works for ALL regular polygons.
Tip 4: Check if your answer makes sense (reasonable size).
Advanced Examples & Practice
4 Pentagon Unknown Side
Exercise 4
A regular pentagon has a perimeter of 25 cm. What is the length of each side?
Definition:

Regular pentagon: A five-sided polygon with all sides and angles equal

Known Perimeter
25 cm
Number of Sides
5 sides
Unknown Side
25 ÷ 5 = 5 cm
Step 1: Identify what you know

Perimeter = 25 cm, Pentagon has 5 equal sides

Step 2: Apply the formula

Side length = 25 ÷ 5 = 5 cm

Step 3: Verify

5 + 5 + 5 + 5 + 5 = 25 cm ✓

Each side = 5 cm
Final answer:

Each side of the pentagon is 5 cm long

Applied rules:

• Division Rule: Side length = Perimeter ÷ Number of sides

• Pentagon Property: Has 5 equal sides

5 Octagon Unknown Side
Exercise 5
A regular octagon has a perimeter of 40 cm. What is the length of each side?
Definition:

Regular octagon: An eight-sided polygon with all sides and angles equal

Known Perimeter
40 cm
Number of Sides
8 sides
Unknown Side
40 ÷ 8 = 5 cm
Step 1: Identify what you know

Perimeter = 40 cm, Octagon has 8 equal sides

Step 2: Apply the formula

Side length = 40 ÷ 8 = 5 cm

Step 3: Verify

5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 40 cm ✓

Each side = 5 cm
Final answer:

Each side of the octagon is 5 cm long

Applied rules:

• Division Rule: Side length = Perimeter ÷ Number of sides

• Octagon Property: Has 8 equal sides

Complete Learning Guide: Finding Unknown Sides
s = P ÷ n
Unknown Side Formula
Key definitions:

Perimeter (P): The total distance around the outside of a shape

Side Length (s): The measurement of one side of the polygon

Number of Sides (n): The count of sides in the polygon

Unknown Side: A side length that needs to be calculated

Regular Polygon: A shape with all sides equal and all angles equal

Complete methodology:
  1. Read the problem carefully: Identify what is given and what you need to find
  2. Count the number of sides: Determine how many sides the polygon has
  3. Set up the equation: Use P = n × s or rearrange to s = P ÷ n
  4. Solve for the unknown: Perform the division to find the side length
  5. Verify your answer: Multiply the side length by the number of sides to check the perimeter
Triangle
s = P ÷ 3
3 sides
Square
s = P ÷ 4
4 sides
Pentagon
s = P ÷ 5
5 sides
Hexagon
s = P ÷ 6
6 sides
Heptagon
s = P ÷ 7
7 sides
Octagon
s = P ÷ 8
8 sides
Tip 1: Count sides carefully - a pentagon has 5, hexagon has 6, octagon has 8.
Tip 2: In regular polygons, all sides are equal, so this method works perfectly.
Tip 3: Always verify by multiplying: Side length × Number of sides = Original perimeter.
Tip 4: Remember the general formula: s = P ÷ n (side = perimeter ÷ number of sides).
Common errors: Dividing by the wrong number, not counting sides correctly, arithmetic mistakes.
Exam preparation: Memorize polygon names, practice division, master the s = P ÷ n formula.
Key formulas to remember:

• General formula: s = P ÷ n

• Triangle: s = P ÷ 3

• Square: s = P ÷ 4

• Pentagon: s = P ÷ 5

• Hexagon: s = P ÷ 6

• Heptagon: s = P ÷ 7

• Octagon: s = P ÷ 8

Detailed Summary: Finding Unknown Sides
Core Concepts:

Perimeter: The total distance around the outside of a shape. It's the sum of all side lengths.

Unknown Side: A side length that is not given in the problem and needs to be calculated.

Regular Polygon: A polygon where all sides are equal in length and all angles are equal.

Division Relationship: When perimeter and number of sides are known, side length = perimeter ÷ number of sides.

Step-by-Step Process:
  1. Identify Given Information: Find the perimeter and determine the type of polygon (number of sides)
  2. Recognize the Pattern: For regular polygons, all sides are equal
  3. Set Up the Equation: Use the relationship P = n × s (where P=perimeter, n=number of sides, s=side length)
  4. Rearrange to Solve: s = P ÷ n
  5. Calculate: Perform the division
  6. Verify: Multiply the answer by the number of sides to ensure it equals the original perimeter
Simple Example
Square with P=12cm
s = 12 ÷ 4 = 3cm
Moderate Example
Pentagon with P=25cm
s = 25 ÷ 5 = 5cm
Advanced Example
Octagon with P=48cm
s = 48 ÷ 8 = 6cm
Memory Trick: "To find one side, divide the total perimeter by the number of sides."
Common Mistake: Don't forget to verify your answer by multiplying back.
Check Reasonableness: Make sure your answer makes sense (not too big or too small).
Pattern Recognition: The more you practice, the faster you'll recognize the division pattern.
Key Principle: Division is the inverse operation of multiplication. Since P = n × s, then s = P ÷ n.
Application: This method only works for regular polygons where all sides are equal.
Essential Takeaways:

• The formula s = P ÷ n is fundamental for finding unknown sides in regular polygons

• Always count the number of sides correctly before dividing

• Verification by multiplication ensures accuracy

• This skill connects perimeter, division, and geometric reasoning

• Understanding this concept prepares you for more complex geometry problems

Questions & Answers

Question: I don't understand why we divide to find the side length. Why don't we subtract or add?

Answer: Great question! This is about understanding the relationship between the parts and the whole:

  • When we calculate perimeter normally: P = n × s (we multiply number of sides by side length)
  • So when we know the perimeter and want to find the side: s = P ÷ n (we divide)
  • Division is the opposite of multiplication

Think of it like sharing cookies equally: If you have 12 cookies to share among 4 friends, each gets 12 ÷ 4 = 3 cookies.

Same idea: If the total perimeter (12) is shared equally among 4 sides, each side gets 12 ÷ 4 = 3 units.

Question: My child is having trouble remembering how many sides different polygons have. How can I help them?

Answer: Here are helpful memory aids for polygon names:

  • Tri-angle: Think "tricycle" (3 wheels) → Triangle has 3 sides
  • Square: Just remember it has 4 equal sides
  • Penta-gon: Think "Pentagon building" in Washington (5 sides)
  • Hexa-gon: "Hex" sounds like "six" → 6 sides
  • Hepta-gon: "Hept" sounds like "heaven" (7 days in a week)
  • Octa-gon: "Octopus" has 8 arms → 8 sides

You can also draw shapes together and count the sides while naming them. Practice regularly with simple games!

Question: How can I help students check their work when finding unknown sides?

Answer: Teach students the verification strategy:

  • Multiply back: Take the found side length and multiply by the number of sides
  • Compare: Does the result equal the original perimeter?
  • Example: If perimeter was 20 and it's a square, found side should be 5. Check: 5 × 4 = 20 ✓

This creates a habit of checking work and builds confidence in their answers.

Also encourage estimation: A reasonable side length should be smaller than the total perimeter.