Perimeter and Area Word Problems PDF - Solved Exercises

Master perimeter and area word problems: rectangles, squares, triangles with 10 detailed exercises and solving methods.

Solution: Exercises 1 to 5
1 Garden Perimeter Problem
Exercise 1
Maria has a rectangular garden that is 8 meters long and 5 meters wide. She wants to put a fence around the entire garden. How many meters of fencing does she need?
Definition:

Perimeter of a rectangle: P = 2 × (length + width) or P = 2l + 2w

Problem Context:

Fencing goes around the outside of the garden, so we need the perimeter.

Perimeter solving method:
  1. Identify the shape: rectangle
  2. Find the dimensions: length = 8m, width = 5m
  3. Apply the perimeter formula: P = 2(l + w)
  4. Calculate and express the answer with units
8×5
Given
l=8, w=5
Formula
P=2(l+w)
Calculate
P=26m
Step 1: Identify the shape and dimensions: Rectangle with length = 8m, width = 5m
Step 2: Apply the perimeter formula: P = 2 × (8 + 5)
Step 3: Calculate: P = 2 × 13 = 26 meters
Fencing needed = 26 meters
Final answer:

Maria needs 26 meters of fencing to surround her garden.

Applied rules:

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

• Real-world context: Fencing goes around perimeter

• Unit consistency: Answer in meters

2 Room Area Problem
Exercise 2
Tom's bedroom is shaped like a square with each side measuring 6 feet. What is the area of his bedroom floor?
Definition:

Area of a square: A = side × side or A = s²

Problem Context:

Area covers the inside of the room, so we need the area.

6×6
Given
s=6 ft
Formula
A=s²
Calculate
A=36 ft²
Step 1: Identify the shape and dimensions: Square with side = 6 feet
Step 2: Apply the area formula: A = 6 × 6
Step 3: Calculate: A = 36 square feet
Area = 36 square feet
Final answer:

The area of Tom's bedroom floor is 36 square feet.

Applied rules:

• Square area formula: A = s²

• Real-world context: Floor area is interior space

• Area units: Square feet (ft²)

3 Tile Perimeter Problem
Exercise 3
A square tile has an area of 25 square inches. What is the perimeter of the tile?
Definition:

Perimeter of a square: P = 4 × side length or P = 4s

Problem Context:

We know the area, need to find the side length first, then perimeter.

?×?
Given
A=25 in²
Find side
s=√25=5
Perimeter
P=20 in
Step 1: Find the side length: Since A = s², then s = √25 = 5 inches
Step 2: Apply perimeter formula: P = 4 × 5
Step 3: Calculate: P = 20 inches
Perimeter = 20 inches
Final answer:

The perimeter of the tile is 20 inches.

Applied rules:

• Square area formula: A = s²

• Square root: To find side from area

• Square perimeter formula: P = 4s

4 Picture Frame Area Problem
Exercise 4
A rectangular picture frame is 10 inches long and 7 inches wide. What is the area of the picture that fits inside the frame?
Definition:

Area of a rectangle: A = length × width or A = lw

Problem Context:

The picture fits inside the frame, so we need the area of the rectangle.

10×7
Given
l=10, w=7
Formula
A=lw
Calculate
A=70 in²
Step 1: Identify the shape and dimensions: Rectangle with length = 10 in, width = 7 in
Step 2: Apply the area formula: A = 10 × 7
Step 3: Calculate: A = 70 square inches
Area = 70 square inches
Final answer:

The area of the picture that fits inside the frame is 70 square inches.

Applied rules:

• Rectangle area formula: A = lw

• Real-world context: Interior space of frame

• Area units: Square inches (in²)

5 Triangular Garden Perimeter
Exercise 5
Sarah wants to put a fence around her triangular flower bed. The sides measure 3 feet, 4 feet, and 5 feet. How much fencing does she need?
Definition:

Perimeter of a triangle: P = a + b + c (sum of all sides)

Problem Context:

Fencing goes around the outside of the triangle, so we need the perimeter.

Sides
3+4+5
Perimeter
P = 12 ft
Step 1: Identify the shape and side lengths: Triangle with sides 3 ft, 4 ft, 5 ft
Step 2: Apply perimeter formula: P = 3 + 4 + 5
Step 3: Calculate: P = 12 feet
Fencing needed = 12 feet
Final answer:

Sarah needs 12 feet of fencing to surround her triangular flower bed.

Applied rules:

• Triangle perimeter formula: P = a + b + c

• Real-world context: Fencing goes around perimeter

• Unit consistency: Answer in feet

Solution: Exercises 6 to 10
6 Playground Area Problem
Exercise 6
A rectangular playground is 20 yards long and 15 yards wide. What is the area of the playground?
Definition:

Area of a rectangle: A = length × width or A = lw

Problem Context:

Area covers the inside of the playground, so we need the area.

20×15
Given
l=20, w=15
Formula
A=lw
Calculate
A=300 yd²
Step 1: Identify the shape and dimensions: Rectangle with length = 20 yd, width = 15 yd
Step 2: Apply the area formula: A = 20 × 15
Step 3: Calculate: A = 300 square yards
Area = 300 square yards
Final answer:

The area of the playground is 300 square yards.

Applied rules:

• Rectangle area formula: A = lw

• Real-world context: Interior space of playground

• Area units: Square yards (yd²)

7 Room Perimeter Problem
Exercise 7
A rectangular room has an area of 48 square feet and is 6 feet wide. What is the perimeter of the room?
Definition:

Rectangle perimeter: P = 2(l + w), where l = A/w

Problem Context:

We know the area and width, need to find length, then perimeter.

?×6
Given
A=48, w=6
Find length
l=A/w=8
Perimeter
P=28 ft
Step 1: Find the length: Since A = lw, then l = A/w = 48/6 = 8 feet
Step 2: Apply perimeter formula: P = 2(8 + 6)
Step 3: Calculate: P = 2 × 14 = 28 feet
Perimeter = 28 feet
Final answer:

The perimeter of the room is 28 feet.

Applied rules:

• Rectangle area formula: A = lw

• Algebraic manipulation: l = A/w

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

8 Square Rug Area Problem
Exercise 8
A square rug has a perimeter of 24 feet. What is the area of the rug?
Definition:

Area of a square: A = s², where s = P/4

Problem Context:

We know the perimeter, need to find the side length, then area.

?×?
Given
P=24 ft
Find side
s=P/4=6
Area
A=36 ft²
Step 1: Find the side length: Since P = 4s, then s = P/4 = 24/4 = 6 feet
Step 2: Apply area formula: A = 6 × 6
Step 3: Calculate: A = 36 square feet
Area = 36 square feet
Final answer:

The area of the rug is 36 square feet.

Applied rules:

• Square perimeter formula: P = 4s

• Algebraic manipulation: s = P/4

• Square area formula: A = s²

9 Pool Deck Perimeter Problem
Exercise 9
A rectangular swimming pool is 12 feet long and 8 feet wide. There is a 2-foot wide deck around the entire pool. What is the perimeter of the outer edge of the deck?
Definition:

Extended rectangle perimeter: Add twice the border width to each dimension

Problem Context:

Deck extends around the pool, so we need the perimeter of the larger rectangle.

16×12
Pool size
12×8
With deck
16×12
Perimeter
P=56 ft
Step 1: Find the outer dimensions: Add 2 feet to each side (2 feet on each side = 4 feet total added to each dimension)
Step 2: New length = 12 + 4 = 16 feet, New width = 8 + 4 = 12 feet
Step 3: Apply perimeter formula: P = 2(16 + 12) = 2 × 28 = 56 feet
Perimeter = 56 feet
Final answer:

The perimeter of the outer edge of the deck is 56 feet.

Applied rules:

• Dimension extension: Add border width to both ends of each dimension

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

• Real-world context: Outer edge of extended area

10 Combined Shapes Problem
Exercise 10
A rectangular garden (10 ft × 6 ft) has a square flower bed (3 ft × 3 ft) in one corner. What is the area of the garden that is NOT covered by the flower bed?
Definition:

Remaining area: Total area minus the area of the subtracted part

Problem Context:

Need to subtract the flower bed area from the total garden area.

10×6
Garden area
60 ft²
Flower bed
9 ft²
Remaining
51 ft²
Step 1: Find the area of the garden: A = 10 × 6 = 60 square feet
Step 2: Find the area of the flower bed: A = 3 × 3 = 9 square feet
Step 3: Subtract to find remaining area: 60 - 9 = 51 square feet
Remaining area = 51 square feet
Final answer:

The area of the garden that is NOT covered by the flower bed is 51 square feet.

Applied rules:

• Area subtraction: Total area - covered area = remaining area

• Rectangle area formula: A = lw

• Square area formula: A = s²

Area and Perimeter Formulas
Perimeter = Distance around shape
Perimeter Concepts
Rectangle
P = 2(l + w)
Perimeter of rectangle
Square
P = 4s
Perimeter of square
Triangle
P = a + b + c
Perimeter of triangle
Area = Space inside shape
Area Concepts
Rectangle
A = l × w
Area of rectangle
Square
A = s²
Area of square
Triangle
A = ½bh
Area of triangle
Perimeter
Distance around
Area
Space inside
Units
Linear (ft, m)
Units
Squared (ft², m²)
Key concept: Perimeter measures distance around, area measures space inside.
Unit difference: Perimeter uses linear units, area uses square units.
Real-world: Fencing = perimeter, flooring = area.
Essential Formulas:

• Rectangle perimeter: P = 2(l + w)

• Rectangle area: A = l × w

• Square perimeter: P = 4s

• Square area: A = s²

• Triangle perimeter: P = a + b + c

Detailed Summary: Perimeter and Area Word Problems
Perimeter vs Area
Core Concepts
Key definitions:

Perimeter: The total distance around the edge of a shape (linear measurement)

Area: The amount of space inside a shape (square measurement)

Word problem: A mathematical problem presented in narrative form

Rectangle: A four-sided shape with four right angles

Square: A rectangle with all sides equal

Triangle: A three-sided polygon

Word problem solving methodology:
  1. Read carefully: Understand what the problem is asking
  2. Identify key information: Find the numbers and units given
  3. Determine shape: Identify whether it's rectangle, square, triangle, etc.
  4. Decide operation: Is it asking for perimeter or area?
  5. Select formula: Choose the correct formula for the shape
  6. Substitute values: Plug the numbers into the formula
  7. Calculate: Perform the mathematical operations
  8. Check units: Ensure answer has correct units
  9. Verify answer: Does it make sense in the context?
Tip 1: Look for clue words: "around" or "fence" means perimeter, "inside" or "cover" means area.
Tip 2: Always include units in your final answer.
Tip 3: Draw a picture to visualize the problem.
Tip 4: Check if your answer is reasonable for the situation.
Tip 5: For combined shapes, break the problem into parts.
Tip 6: When extending dimensions, add the extension to both sides.
Common errors: Mixing up perimeter and area, forgetting to square units for area, calculation mistakes.
Problem-solving strategy: Read twice, draw a picture, identify the question, choose the right formula.
Key Formulas to Remember:

• Rectangle perimeter: P = 2(l + w)

• Rectangle area: A = l × w

• Square perimeter: P = 4s

• Square area: A = s²

• Triangle perimeter: P = a + b + c

Simple Examples

Rectangle: l=5, w=3 → P = 2(5+3) = 16, A = 5×3 = 15

Square: s=4 → P = 4×4 = 16, A = 4² = 16

Triangle: sides 3,4,5 → P = 3+4+5 = 12

Advanced Examples

Find missing dimension: Rectangle A=24, w=4, find l → l = A/w = 24/4 = 6

Extension problem: Original 10×6 with 2-unit border → New: 14×10

Combined areas: Total area - covered area = remaining area

P = 2(l + w) for rectangles
A = l × w for rectangles
Fundamental Formulas

Questions & Answers

Question: How do I know if a word problem is asking for perimeter or area? They seem similar.

Answer: Great question! Here's how to tell the difference:

  • Perimeter: Questions about "around," "fence," "border," "edge," "outline," or "distance around"
  • Area: Questions about "inside," "cover," "floor," "surface," "space," or "fill"

Think of it this way: If you're walking around the edge of something, you're measuring perimeter. If you're covering the inside with paint or carpet, you're measuring area. In the problems above, fencing needs perimeter while flooring needs area!

Question: My child is struggling with word problems. How can I help them break down these problems?

Answer: Here's a step-by-step approach to help your child:

  • Read aloud: Have them read the problem out loud to process it
  • Circle key numbers: Identify all the important measurements
  • Draw a picture: Visual representation helps clarify the problem
  • Label the drawing: Put the numbers in the right places
  • Ask: perimeter or area?: Identify what the question is asking
  • Write the formula: Write down the appropriate formula
  • Solve step by step: Work through the calculation

Practice with simple problems first, then gradually increase difficulty. Consistency is key!

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

Answer: Here are memory tricks for the formulas:

  • Rectangle perimeter: "Two lengths plus two widths" or "2 times (length + width)"
  • Rectangle area: "Length times width" - think of tiles filling the space
  • Square perimeter: "Four sides the same" - 4 times one side
  • Square area: "Side times itself" - imagine a square grid
  • Triangle perimeter: "All three sides added together"

Practice drawing the shapes and writing the formulas. The more you practice, the more automatic it becomes!