Complete Guide to Perimeter and Area of the Polygon for 3rd Grade

Master calculating perimeter and area of rectangles, squares, and polygons with detailed exercises and solutions.

Perimeter Exercises
1 Rectangle perimeter
Exercise 1
A rectangle has a length of 10 cm and a width of 6 cm. What is its perimeter?
Definition:

Perimeter: The total distance around the outside of a shape. For a rectangle: P = 2 × (length + width)

Method:
  1. Write the perimeter formula for a rectangle
  2. Substitute known values
  3. Calculate the result
  4. Include the unit
Given
l = 10, w = 6
Formula
P = 2(l+w)
Calculate
P = 32 cm
Step 1: Write the perimeter formula

For a rectangle: Perimeter = 2 × (length + width)

P = 2(l + w)

Step 2: Substitute known values

P = 2(10 + 6)

Step 3: Calculate

P = 2(16) = 32

Step 4: Include unit

The perimeter is 32 cm.

Perimeter = 32 cm
Final answer:

The perimeter of the rectangle is 32 cm.

Applied rules:

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

• Addition: Add length and width first

• Multiplication: Multiply by 2

2 Square perimeter
Exercise 2
A square has sides of 7 cm. What is its perimeter?
Definition:

Square Perimeter: Since all sides are equal, P = 4 × side length

Given
side = 7 cm
Formula
P = 4s
Calculate
P = 28 cm
Step 1: Write the perimeter formula for a square

For a square: Perimeter = 4 × side length

P = 4s

Step 2: Substitute the side length

P = 4(7)

Step 3: Calculate

P = 28

Step 4: Include unit

The perimeter is 28 cm.

Perimeter = 28 cm
Final answer:

The perimeter of the square is 28 cm.

Applied rules:

• Square perimeter formula: P = 4s

• All sides equal: In a square, all four sides are the same length

• Multiplication: Multiply side length by 4

3 Triangle perimeter
Exercise 3
A triangle has sides of 5 cm, 7 cm, and 9 cm. What is its perimeter?
Definition:

Triangle Perimeter: Sum of all three sides. P = a + b + c

Given
a = 5, b = 7, c = 9
Formula
P = a + b + c
Calculate
P = 21 cm
Step 1: Write the perimeter formula for a triangle

For a triangle: Perimeter = side₁ + side₂ + side₃

P = a + b + c

Step 2: Substitute known values

P = 5 + 7 + 9

Step 3: Calculate

P = 21

Step 4: Include unit

The perimeter is 21 cm.

Perimeter = 21 cm
Final answer:

The perimeter of the triangle is 21 cm.

Applied rules:

• Triangle perimeter formula: Sum of all three sides

• Addition: Add all three side lengths

• Verification: Check that you added all sides

Area Exercises
▭
Rectangle
P = 2(l + w)

Perimeter

A = l × w

Area

□
Square
P = 4s

Perimeter

A = s²

Area

△
Triangle
P = a + b + c

Perimeter

A = ½bh

Area

⬟
Polygon
P = sum of sides

Perimeter

Area varies

Area: A = length × width
Area of Rectangle
Rectangle
P = 2(l + w)
Perimeter
Rectangle
A = l × w
Area
Square
P = 4s
Perimeter
Square
A = s²
Area
4 Rectangle area
Exercise 4
A rectangle has a length of 8 cm and a width of 5 cm. What is its area?
Definition:

Area: The amount of space inside a shape. For a rectangle: A = length × width

Given
l = 8, w = 5
Formula
A = l × w
Calculate
A = 40 cm²
Step 1: Write the area formula for a rectangle

For a rectangle: Area = length × width

A = l × w

Step 2: Substitute known values

A = 8 × 5

Step 3: Calculate

A = 40

Step 4: Include unit

The area is 40 cm² (square centimeters).

Area = 40 cm²
Final answer:

The area of the rectangle is 40 cm².

5 Square area
Exercise 5
A square has sides of 6 cm. What is its area?
Definition:

Square Area: Since all sides are equal, A = side × side = s²

Given
side = 6 cm
Formula
A = s²
Calculate
A = 36 cm²
Step 1: Write the area formula for a square

For a square: Area = side × side = side²

A = s²

Step 2: Substitute the side length

A = 6²

Step 3: Calculate

A = 6 × 6 = 36

Step 4: Include unit

The area is 36 cm² (square centimeters).

Area = 36 cm²
Final answer:

The area of the square is 36 cm².

Polygon Exercises
6 Pentagon perimeter
Exercise 6
A regular pentagon has sides of 4 cm. What is its perimeter?
Definition:

Regular Pentagon: A five-sided shape with all sides equal. P = 5 × side length.

Given
side = 4 cm
Formula
P = 5s
Calculate
P = 20 cm
Step 1: Write the perimeter formula for a regular pentagon

For a regular pentagon: Perimeter = 5 × side length

P = 5s

Step 2: Substitute the side length

P = 5(4)

Step 3: Calculate

P = 20

Step 4: Include unit

The perimeter is 20 cm.

Perimeter = 20 cm
Final answer:

The perimeter of the pentagon is 20 cm.

7 Hexagon perimeter
Exercise 7
A regular hexagon has sides of 5 cm. What is its perimeter?
Definition:

Regular Hexagon: A six-sided shape with all sides equal. P = 6 × side length.

Given
side = 5 cm
Formula
P = 6s
Calculate
P = 30 cm
Step 1: Write the perimeter formula for a regular hexagon

For a regular hexagon: Perimeter = 6 × side length

P = 6s

Step 2: Substitute the side length

P = 6(5)

Step 3: Calculate

P = 30

Step 4: Include unit

The perimeter is 30 cm.

Perimeter = 30 cm
Final answer:

The perimeter of the hexagon is 30 cm.

Complete Summary: Perimeter and Area of the Polygon

Key Concepts and Definitions:

Perimeter: The total distance around the outside of a 2D shape. It's like walking all the way around the edge of the shape.

Area: The amount of space inside a 2D shape. It's like covering the entire surface with squares.

Polygon: Any shape made up of straight lines. Examples: triangle, rectangle, square, pentagon, hexagon.

Rectangle: A 4-sided shape where opposite sides are equal and all angles are 90°.

Square: A special rectangle where all 4 sides are equal.

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

Core Rules and Principles:

• Perimeter Rule: Perimeter = sum of all side lengths

• Rectangle Perimeter: P = 2(length + width)

• Square Perimeter: P = 4 × side length

• Triangle Perimeter: P = side₁ + side₂ + side₃

• Rectangle Area: A = length × width

• Square Area: A = side²

• Regular Polygon Perimeter: P = number of sides × side length

Step-by-Step Method for Perimeter:
  1. Identify the shape: Determine if it's a rectangle, square, triangle, etc.
  2. Count the sides: Know how many sides the shape has
  3. Measure or get side lengths: Find the length of each side
  4. Add all side lengths: Sum them up to get the perimeter
  5. State the answer: Include the unit (cm, m, inches, etc.)
Step-by-Step Method for Area:
  1. Identify the shape: Determine if it's a rectangle, square, etc.
  2. Select the formula: Use the correct area formula
  3. Substitute values: Put the measurements into the formula
  4. Calculate: Do the multiplication
  5. State the answer: Include the unit (cm², m², etc.)
Perimeter Example: Rectangle

Rectangle with length = 6 cm and width = 4 cm

Step 1: P = 2(l + w) = 2(6 + 4) = 2(10) = 20 cm

Step 2: The perimeter is 20 cm

Area Example: Rectangle

Rectangle with length = 6 cm and width = 4 cm

Step 1: A = l × w = 6 × 4 = 24 cm²

Step 2: The area is 24 cm²

Perimeter Example: Square

Square with side = 5 cm

Step 1: P = 4s = 4(5) = 20 cm

Step 2: The perimeter is 20 cm

Area Example: Square

Square with side = 5 cm

Step 1: A = s² = 5² = 25 cm²

Step 2: The area is 25 cm²

Perimeter Example: Regular Pentagon

Regular pentagon with side = 3 cm

Step 1: P = 5s = 5(3) = 15 cm

Step 2: The perimeter is 15 cm

Memory Tip 1: "Perimeter" sounds like "fence-meter" - you measure the fence around a yard!
Memory Tip 2: "Area" sounds like "are-a" - it covers the whole area inside!
Memory Tip 3: Perimeter = outside edge, Area = inside space.
Common Mistake: Don't confuse perimeter (addition) with area (multiplication).
Common Mistake: Always include units in your answer (cm, m, cm², m², etc.)
Key Takeaways: 1) Perimeter measures the outside edge, 2) Area measures the inside space, 3) Use different formulas for each.
Real World Application: Perimeter for fencing, area for carpeting or painting.
Perimeter and Area Formula Reference
P = sum of all sides
Perimeter Formula
A = length × width
Area Formula
Perimeter Formulas:

Rectangle: P = 2(l + w) where l = length, w = width

Square: P = 4s where s = side length

Triangle: P = a + b + c where a, b, c are side lengths

Regular Pentagon: P = 5s where s = side length

Regular Hexagon: P = 6s where s = side length

Regular Polygon: P = n × s where n = number of sides, s = side length

Area Formulas:

Rectangle: A = l × w where l = length, w = width

Square: A = s² where s = side length

Complete methodology:
  1. Identify the shape: Recognize whether it's a rectangle, square, triangle, etc.
  2. Select the formula: Use the appropriate perimeter or area formula
  3. Substitute known values: Fill in the given measurements
  4. Calculate: Perform the necessary arithmetic operations
  5. Verify solution: Check that your answer makes sense
Tip 1: Always label your sides clearly to avoid confusion.
Tip 2: Remember that rectangles have 2 pairs of equal sides.
Tip 3: Draw the shape and label known sides to visualize the problem.
Tip 4: Always check your answer by calculating the perimeter/area with your solution.
Common errors: Forgetting that rectangles have 2 lengths and 2 widths, mixing up perimeter and area formulas, arithmetic mistakes.
Key formulas: Rectangle: P=2(l+w), A=l×w, Square: P=4s, A=s²
Essential rules to remember:

• Rectangle rule: P = 2(length + width), A = length × width

• Square rule: P = 4 × side length, A = side²

• Triangle rule: P = side₁ + side₂ + side₃

• Regular polygon rule: P = number of sides × side length

• Units: Perimeter in cm, m, inches; Area in cm², m², etc.

Examples: P=2(5+3)=16, A=5×3=15, P=4(5)=20, A=5²=25
Formula Applications

Questions & Answers

Question: I'm confused about why we multiply by 2 in the rectangle perimeter formula. Why isn't it just length + width?

Answer: Great question! When we calculate perimeter, we need to count ALL the sides of the rectangle. A rectangle has 4 sides:

  • One length side
  • One width side
  • Another length side (opposite to the first)
  • Another width side (opposite to the first)

So we have: length + width + length + width = 2×length + 2×width = 2(length + width)

Think of walking around the rectangle - you travel the length twice and the width twice!

Example: If length = 5 and width = 3, the perimeter is 5+3+5+3 = 16, which equals 2(5+3) = 16.

Question: My child is struggling with the difference between perimeter and area. How can I help them understand?

Answer: Here's a helpful way to explain the difference:

  1. Perimeter: Think of it as building a fence around a yard - you measure the distance around the outside
  2. Area: Think of it as covering the floor with tiles - you measure the space inside

Use physical examples: If you have a rectangular piece of paper:

  • Perimeter = measure the tape needed to go around the edge
  • Area = count how many small squares fit inside

Perimeter is measured in units (cm, m), while area is measured in square units (cm², m²).

Question: Are there any memory tricks to remember the different perimeter and area formulas?

Answer: Yes! Here are some effective memory tricks:

  • Perimeter: "Perimeter = Peri-Meter (around-measure)" - it goes around the shape
  • Area: "Area = Are-A (the space inside)"
  • Rectangle Perimeter: "Two lengths, two widths" → P = 2(l + w)
  • Square Perimeter: "Four equal sides" → P = 4s
  • Rectangle Area: "Length times width" → A = l × w
  • Square Area: "Side times side" → A = s²

Physical activity: Walk around the perimeter of different shaped objects while counting the sides out loud, and trace the area with your hand to cover the space inside.