How to Calculate the Perimeter of a Polygon - Solved Exercises

Master polygon perimeter calculation: triangles, rectangles, squares, pentagons, hexagons with 10 detailed exercises and visual examples.

Solution: Exercises 1 to 5
1 Triangle Perimeter
Exercise 1
Calculate the perimeter of a triangle with sides measuring 5 cm, 7 cm, and 9 cm.
Definition:

Perimeter of a polygon: The sum of all side lengths of the shape

Perimeter calculation method:
  1. Add up all the side lengths of the polygon
  2. Make sure all measurements are in the same unit
  3. Express the answer with the appropriate unit
Triangle sides
5 + 7 + 9
Perimeter
21 cm
Step 1: Identify all side lengths

The triangle has sides of 5 cm, 7 cm, and 9 cm

Step 2: Add all side lengths together

Perimeter = 5 + 7 + 9 = 21 cm

Step 3: Express with correct unit

The perimeter is 21 cm

Perimeter = 21 cm
Final answer:

The perimeter of the triangle is 21 cm

Applied rules:

• Perimeter rule: Sum of all side lengths

• Unit consistency: All measurements in centimeters

• Addition: Basic arithmetic operation

2 Rectangle Perimeter
Exercise 2
Calculate the perimeter of a rectangle with length 8 cm and width 4 cm.
Definition:

Rectangle perimeter: P = 2 × (length + width) or P = 2l + 2w

8×4
Formula
P = 2(l + w)
Substitute
P = 2(8 + 4)
Calculate
P = 24 cm
Step 1: Identify length and width

Length = 8 cm, Width = 4 cm

Step 2: Apply rectangle perimeter formula

P = 2 × (length + width) = 2 × (8 + 4)

Step 3: Calculate the result

P = 2 × 12 = 24 cm

Perimeter = 24 cm
Final answer:

The perimeter of the rectangle is 24 cm

Applied rules:

• Rectangle perimeter formula: P = 2(l + w)

• Order of operations: Parentheses first

• Multiplication: Multiply by 2 after addition

3 Square Perimeter
Exercise 3
Calculate the perimeter of a square with side length 6 cm.
Definition:

Square perimeter: P = 4 × side length or P = 4s

6×6
Square side
6 cm
Formula
P = 4s
Result
P = 24 cm
Step 1: Identify the side length

All sides of a square are equal: 6 cm each

Step 2: Apply square perimeter formula

P = 4 × side length = 4 × 6

Step 3: Calculate the result

P = 24 cm

Perimeter = 24 cm
Final answer:

The perimeter of the square is 24 cm

Applied rules:

• Square perimeter formula: P = 4s

• Square property: All sides equal

• Multiplication: Multiply side length by 4

4 Pentagon Perimeter
Exercise 4
Calculate the perimeter of a pentagon with sides measuring 3 cm, 4 cm, 5 cm, 6 cm, and 7 cm.
Definition:

Polygon perimeter: Sum of all side lengths regardless of number of sides

5 sides
Side lengths
3+4+5+6+7
Perimeter
P = 25 cm
Step 1: List all side lengths

The pentagon has sides: 3 cm, 4 cm, 5 cm, 6 cm, 7 cm

Step 2: Add all side lengths

P = 3 + 4 + 5 + 6 + 7 = 25 cm

Step 3: Express with correct unit

The perimeter is 25 cm

Perimeter = 25 cm
Final answer:

The perimeter of the pentagon is 25 cm

Applied rules:

• Perimeter rule: Sum of all side lengths

• Addition: Basic arithmetic operation

• Unit consistency: All measurements in centimeters

5 Hexagon Perimeter
Exercise 5
Calculate the perimeter of a regular hexagon with each side measuring 4 cm.
Definition:

Regular polygon perimeter: P = n × side length (where n is number of sides)

6 sides
Hexagon sides
6 sides
Side length
4 cm
Perimeter
P = 24 cm
Step 1: Identify number of sides and side length

Hexagon has 6 sides, each measuring 4 cm

Step 2: Apply regular polygon formula

P = number of sides × side length = 6 × 4

Step 3: Calculate the result

P = 24 cm

Perimeter = 24 cm
Final answer:

The perimeter of the hexagon is 24 cm

Applied rules:

• Regular polygon formula: P = n × s

• Hexagon property: 6 equal sides

• Multiplication: Multiply number of sides by side length

Solution: Exercises 6 to 10
6 Irregular Quadrilateral Perimeter
Exercise 6
Find the perimeter of a quadrilateral with sides measuring 12 cm, 8 cm, 15 cm, and 10 cm.
Definition:

Irregular polygon perimeter: Still the sum of all side lengths

4 sides
Side lengths
12+8+15+10
Perimeter
P = 45 cm
Step 1: List all side lengths

The quadrilateral has sides: 12 cm, 8 cm, 15 cm, 10 cm

Step 2: Add all side lengths

P = 12 + 8 + 15 + 10 = 45 cm

Step 3: Express with correct unit

The perimeter is 45 cm

Perimeter = 45 cm
Final answer:

The perimeter of the quadrilateral is 45 cm

Applied rules:

• Perimeter rule: Sum of all side lengths

• Addition: Basic arithmetic operation

• Unit consistency: All measurements in centimeters

7 Missing Side Problem
Exercise 7
A rectangle has a perimeter of 30 cm and a length of 9 cm. Find the width.
Definition:

Rectangle perimeter formula: P = 2(l + w), so w = (P/2) - l

9×?
Given info
P=30, l=9
Formula
w=(P/2)-l
Calculate
w = 6 cm
Step 1: Write the perimeter formula

P = 2(l + w), where P = 30 cm and l = 9 cm

Step 2: Substitute known values

30 = 2(9 + w)

Step 3: Solve for width

30/2 = 9 + w → 15 = 9 + w → w = 6 cm

Width = 6 cm
Final answer:

The width of the rectangle is 6 cm

Applied rules:

• Rectangle perimeter formula: P = 2(l + w)

• Algebraic manipulation: Isolate unknown variable

• Arithmetic: Division and subtraction operations

8 Square Side Length
Exercise 8
If a square has a perimeter of 32 cm, find the length of one side.
Definition:

Square side formula: s = P ÷ 4, since P = 4s

?×?
Given
P = 32 cm
Formula
s = P ÷ 4
Calculate
s = 8 cm
Step 1: Write the square perimeter formula

P = 4s, where P = 32 cm

Step 2: Solve for side length

s = P ÷ 4 = 32 ÷ 4

Step 3: Calculate the result

s = 8 cm

Side length = 8 cm
Final answer:

The length of one side of the square is 8 cm

Applied rules:

• Square perimeter formula: P = 4s

• Division: Divide perimeter by 4

• Algebraic manipulation: Isolate the variable

9 Complex Polygon Perimeter
Exercise 9
Calculate the perimeter of a polygon with sides: 2.5 cm, 3.7 cm, 4.1 cm, 2.9 cm, 3.8 cm.
Definition:

Decimal perimeter: Same principle, but with decimal numbers

5 sides
Sides
2.5+3.7+4.1+2.9+3.8
Perimeter
P = 17.0 cm
Step 1: List all side lengths

The polygon has sides: 2.5 cm, 3.7 cm, 4.1 cm, 2.9 cm, 3.8 cm

Step 2: Add all side lengths

P = 2.5 + 3.7 + 4.1 + 2.9 + 3.8 = 17.0 cm

Step 3: Express with correct unit

The perimeter is 17.0 cm

Perimeter = 17.0 cm
Final answer:

The perimeter of the polygon is 17.0 cm

Applied rules:

• Perimeter rule: Sum of all side lengths

• Decimal addition: Align decimal points

• Unit consistency: All measurements in centimeters

10 Real-world Application
Exercise 10
A rectangular garden has a length of 15 meters and a width of 10 meters. How much fencing is needed to enclose the garden?
Definition:

Fencing problem: Fencing needed equals the perimeter of the rectangle

15×10
Dimensions
15m × 10m
Formula
P = 2(l+w)
Fencing
P = 50 m
Step 1: Identify the shape and dimensions

Rectangular garden: length = 15 m, width = 10 m

Step 2: Apply rectangle perimeter formula

P = 2 × (length + width) = 2 × (15 + 10)

Step 3: Calculate the result

P = 2 × 25 = 50 meters

Fencing needed = 50 m
Final answer:

50 meters of fencing is needed to enclose the garden

Applied rules:

• Real-world application: Perimeter represents boundary length

• Rectangle perimeter formula: P = 2(l + w)

• Unit consistency: All measurements in meters

Rules and methods, laws,...
Perimeter = Sum of all sides
Basic Perimeter Rule
Triangle
P = a + b + c
Three sides added together
Rectangle
P = 2(l + w)
Twice the sum of length and width
Square
P = 4s
Four times the side length
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
P = s₁ + s₂ + s₃ + ... + sₙ
General Polygon Formula
Key concept: Perimeter measures the distance around a shape.
Unit: Always express perimeter in linear units (cm, m, km, etc.).
Regular polygons: All sides equal, so multiply side by number of sides.
Essential Perimeter Rules:

• Basic rule: P = sum of all side lengths

• Triangle: P = a + b + c

• Rectangle: P = 2(l + w)

• Square: P = 4s

• Regular n-gon: P = n × s

Polygon Perimeter Fundamentals & Properties
Perimeter = Sum of all sides
Core Concept
Key definitions:

Perimeter: The total distance around the edge of a shape

Polygon: A closed shape made of straight lines

Regular polygon: A polygon with all sides and angles equal

Perimeter calculation methodology:
  1. Identify the shape: Determine if it's a triangle, rectangle, square, or other polygon
  2. Measure or identify side lengths: Get the length of each side
  3. Apply the appropriate formula: Use specific formula or sum all sides
  4. Calculate and check: Add values and verify with correct units
Tip 1: Always check that all measurements are in the same unit before adding.
Tip 2: For regular polygons, multiply the number of sides by the length of one side.
Tip 3: Count the number of sides to make sure you've included all in your sum.
Tip 4: Draw the shape and label sides to visualize the problem better.
Common errors: Forgetting to add all sides, mixing measurement units, miscounting sides.
Exam preparation: Practice with different polygon types, focus on unit consistency.
Perimeter Formulas to Know:

• Any polygon: P = s₁ + s₂ + s₃ + ... + sₙ

• Triangle: P = a + b + c

• Rectangle: P = 2(l + w)

• Square: P = 4s

• Regular n-gon: P = n × s

Questions & Answers

Question: Why do we need to learn about perimeters? When will I ever use this in real life?

Answer: Great question! Perimeter calculations are used in many everyday situations:

  • Fencing: How much fence do you need for your backyard? That's the perimeter!
  • Picture frames: The frame goes around the edge - that's perimeter
  • Garden borders: Measuring how much material for edging
  • Construction: Architects use perimeter to plan room layouts

Understanding perimeter helps you solve practical problems like planning a garden, building a playground, or even arranging furniture in your room. It's a fundamental skill that builds the foundation for more advanced math concepts!

Question: My child is struggling with perimeter calculations. How can I help them understand it better?

Answer: Here are effective strategies to help your child understand perimeter:

  • Hands-on activities: Have them measure the perimeter of books, tables, or rooms using rulers or measuring tapes
  • Visual aids: Draw shapes and trace the edges to show "distance around"
  • Real examples: Walk around the edge of your yard and explain that's the perimeter
  • Practice regularly: Start with simple shapes and gradually increase complexity

Remember, the key concept is "distance around the edge." Once they grasp that, the calculations become much easier. Encourage them to count sides and add carefully!

Question: I sometimes forget which formula to use for different shapes. How can I remember them?

Answer: Here are memory tricks for common perimeter formulas:

  • Any polygon: Remember "Perimeter = All sides added together"
  • Rectangle: Think "2 lengths + 2 widths" or "2(length + width)"
  • Square: "4 times one side" because all sides are equal
  • Triangle: "Side + side + side" - just add all three sides

The most important thing to remember is that perimeter is always the sum of all side lengths. For regular shapes, you can use shortcuts (like 4s for squares), but the basic principle is always adding all sides!