Perimeter, Area and Volume Word Problems - Solved Exercises

Complete exercises with 10 perimeter, area, and volume word problems: rectangles, squares, triangles, cubes, rectangular prisms with detailed solutions and 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 Cube Volume Problem
Exercise 3
A cube-shaped toy box has sides that measure 4 inches each. What is the volume of the toy box?
Definition:

Volume of a cube: V = side × side × side or V = s³

Problem Context:

Volume measures the space inside a 3D shape, so we need the volume.

4×4×4
Given
s=4 in
Formula
V=s³
Calculate
V=64 in³
Step 1: Identify the shape and dimensions: Cube with side = 4 inches
Step 2: Apply the volume formula: V = 4 × 4 × 4
Step 3: Calculate: V = 64 cubic inches
Volume = 64 cubic inches
Final answer:

The volume of the toy box is 64 cubic inches.

Applied rules:

• Cube volume formula: V = s³

• Real-world context: Volume measures space inside

• Volume units: Cubic inches (in³)

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 Rectangular Prism Volume Problem
Exercise 5
A rectangular storage box is 5 inches long, 3 inches wide, and 2 inches tall. What is the volume of the box?
Definition:

Volume of a rectangular prism: V = length × width × height or V = lwh

Problem Context:

Volume measures the space inside the 3D box, so we need the volume.

5×3×2
Given
l=5, w=3, h=2
Formula
V=lwh
Calculate
V=30 in³
Step 1: Identify the shape and dimensions: Rectangular prism with length = 5 in, width = 3 in, height = 2 in
Step 2: Apply the volume formula: V = 5 × 3 × 2
Step 3: Calculate: V = 30 cubic inches
Volume = 30 cubic inches
Final answer:

The volume of the box is 30 cubic inches.

Applied rules:

• Rectangular prism volume formula: V = lwh

• Real-world context: Volume measures space inside

• Volume units: Cubic inches (in³)

Solution: Exercises 6 to 10
6 Triangular Garden Perimeter
Exercise 6
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

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 Cube Storage Problem
Exercise 9
A cube-shaped container has a volume of 125 cubic inches. What is the length of each side of the cube?
Definition:

Side length of a cube: s = ∛V (cube root of volume)

Problem Context:

We know the volume, need to find the side length of the cube.

?×?×?
Given
V=125 in³
Find side
s=∛125=5
Side length
s=5 in
Step 1: We know that V = s³, so s = ∛V
Step 2: Find the cube root: s = ∛125 = 5 inches
Step 3: Verify: 5³ = 5 × 5 × 5 = 125 cubic inches ✓
Side length = 5 inches
Final answer:

The length of each side of the cube is 5 inches.

Applied rules:

• Cube volume formula: V = s³

• Algebraic manipulation: s = ∛V

• Cube root: Finding the inverse of cubing

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

Formulas and Properties
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
Volume = Space inside 3D shape
Volume Concepts
Cube
V = s³
Volume of cube
Rect. Prism
V = lwh
Volume of rectangular prism
Perimeter
Distance around
Area
Space inside
Volume
Space inside 3D
Units
Linear (ft, m)
Units
Squared (ft², m²)
Units
Cubic (ft³, m³)
Key concept: Perimeter measures distance around, area measures space inside, volume measures space in 3D.
Unit differences: Perimeter = linear, Area = square, Volume = cubic.
Real-world: Fencing = perimeter, flooring = area, storage = volume.
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

• Cube volume: V = s³

• Rectangular prism volume: V = lwh

Detailed Summary: Perimeter, Area and Volume Word Problems
2D vs 3D Measurements
Core Concepts
Key definitions:

Perimeter: The total distance around the edge of a 2D shape (linear measurement)

Area: The amount of space inside a 2D shape (square measurement)

Volume: The amount of space inside a 3D shape (cubic 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

Cube: A 3D shape with six square faces

Rectangular prism: A 3D shape with six rectangular faces

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 2D or 3D, rectangle, square, triangle, cube, etc.
  4. Decide operation: Is it asking for perimeter, area, or volume?
  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 (linear, square, or cubic)
  9. Verify answer: Does it make sense in the context?
Tip 1: Look for clue words: "around" or "fence" = perimeter, "inside" or "cover" = area, "space inside" = volume.
Tip 2: Always include units in your final answer (ft, ft², ft³).
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: Remember: Perimeter = linear (1D), Area = square (2D), Volume = cubic (3D).
Common errors: Mixing up perimeter, area, and volume; forgetting to cube units for volume; 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

• Cube volume: V = s³

• Rectangular prism volume: V = lwh

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

Cube: s=3 → V = 3³ = 27

Advanced Examples

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

Find cube side: V=125 → s = ∛125 = 5

Combined areas: Total area - covered area = remaining area

P = 2(l + w) for rectangles
A = l × w for rectangles
V = lwh for rectangular prisms
Fundamental Formulas

Questions & Answers

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

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

  • Perimeter: Questions about "around," "fence," "border," "edge," "outline," or "distance around" (2D)
  • Area: Questions about "inside," "cover," "floor," "surface," "space," or "fill" (2D)
  • Volume: Questions about "space inside," "how much fits," "capacity," or "cubic space" (3D)

Think of it this way: Perimeter is the fence around a yard, area is the grass inside the yard, and volume is the space inside a box!

Question: My child is struggling with volume problems. How can I help them understand the concept?

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

  • Hands-on activities: Use small cubes to build rectangular prisms and count the total cubes
  • Real examples: Fill containers with water or rice to demonstrate capacity
  • Visual aids: Draw 3D shapes and show how length × width × height fills the space
  • Practice regularly: Start with simple cubes and gradually move to rectangular prisms

Remember, volume measures the space inside a 3D object. It's like asking "how many unit cubes fit inside?" The key is helping children visualize the 3-dimensional aspect!

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
  • Cube volume: "Side times itself times itself" - multiply the side 3 times
  • Rectangular prism volume: "Length times width times height" - multiply all three dimensions

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