Solved Exercises on Geometry Word Problems in Grade 7

Master geometry word problems: area, perimeter, volume, and composite shapes through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Perimeter Word Problem
Exercise 1
A rectangular garden has a length that is 3 meters longer than its width. If the perimeter is 54 meters, find the dimensions of the garden.
Visual: 🌿 Garden with unknown width and length = width + 3
Definition:

Perimeter: The distance around a shape

Rectangle Perimeter: P = 2(length + width)

Problem-solving method:
  1. Read the problem carefully and identify known information
  2. Define variables for unknown quantities
  3. Set up an equation based on the given relationship
  4. Solve the equation
  5. Check the solution
Define Variables
w = width, l = w + 3
Set Up Equation
2(w + w + 3) = 54
Solve
w = 12, l = 15
Step 1: Define variables

Let w = width of the garden

Then length = w + 3 (since it's 3 meters longer than width)

Step 2: Set up the equation

Perimeter = 2(length + width)

54 = 2((w + 3) + w)

54 = 2(2w + 3)

Step 3: Solve the equation

54 = 4w + 6

48 = 4w

w = 12 meters

Length = w + 3 = 12 + 3 = 15 meters

Width = 12m, Length = 15m
Final answer:

The garden is 12 meters wide and 15 meters long.

Applied rules:

• Perimeter Formula: P = 2(l + w)

• Algebra: Solving linear equations

• Problem-Solving: Define variables, set up equation

2 Area Word Problem
Exercise 2
A square room has a floor area of 144 square feet. What is the length of one side of the room? How much trim would be needed to go around the entire room?
Visual: 🏠 Square room with equal sides and area = 144 ft²
Definition:

Area: The amount of space inside a shape

Square Area: A = s²

Square Perimeter: P = 4s

Area Formula
s² = 144
Solve for s
s = √144 = 12
Perimeter
P = 4 × 12 = 48
Step 1: Find the side length

Area of square = s² = 144

s = √144 = 12 feet

Step 2: Calculate perimeter

Perimeter of square = 4s

P = 4 × 12 = 48 feet

Step 3: Answer both questions

Side length = 12 feet

Trim needed = 48 feet

Side = 12ft, Trim = 48ft
Final answer:

The side length is 12 feet and 48 feet of trim is needed.

Applied rules:

• Square Area: A = s²

• Square Perimeter: P = 4s

• Square Root: Finding side from area

3 Volume Word Problem
Exercise 3
A rectangular box has dimensions of 8 inches by 6 inches by 4 inches. If the height is increased by 2 inches, what is the new volume? How much did the volume increase?
Visual: 📦 Box with L=8, W=6, H=4 then H=6 after increase
Definition:

Volume: The amount of space inside a 3D shape

Rectangular Prism Volume: V = L × W × H

Original Volume
V₁ = 8 × 6 × 4 = 192
New Volume
V₂ = 8 × 6 × 6 = 288
Increase
ΔV = 288 - 192 = 96
Step 1: Calculate original volume

V₁ = L × W × H = 8 × 6 × 4 = 192 cubic inches

Step 2: Calculate new dimensions

New height = 4 + 2 = 6 inches

V₂ = 8 × 6 × 6 = 288 cubic inches

Step 3: Find the volume increase

ΔV = V₂ - V₁ = 288 - 192 = 96 cubic inches

New Volume = 288in³, Increase = 96in³
Final answer:

The new volume is 288 cubic inches and the volume increased by 96 cubic inches.

Applied rules:

• Volume Formula: V = L × W × H

• Subtraction: Finding differences

• Multiplication: Calculating volumes

Geometry Word Problem Solving Guide
P = 2(l + w)
Rectangle Perimeter
A = l × w
Rectangle Area
V = l × w × h
Rectangular Prism Volume
Square
A = s², P = 4s
All sides equal
Triangle
A = ½bh
Base × height ÷ 2
Circle
A = πr², C = 2πr
Using radius r
Key definitions:

Word Problem: A mathematical problem presented in a narrative format

Perimeter: Distance around the outside of a shape

Area: Amount of space inside a 2D shape

Volume: Amount of space inside a 3D shape

5-Step Problem-Solving Method:
  1. READ: Carefully read the problem multiple times
  2. IDENTIFY: Find what you're asked to find
  3. REPRESENT: Assign variables to unknowns
  4. EQUATE: Write equations based on relationships
  5. SOLVE: Solve equations and check your answer
Tip 1: Draw a diagram to visualize the problem.
Tip 2: Write down all given information before solving.
Tip 3: Check if your answer makes sense in context.
Tip 4: Always include units in your final answer.
Common errors: Misreading the problem, using wrong formulas, forgetting units, arithmetic mistakes.
Key strategies: Define variables clearly, check units, verify solutions.
Essential formulas to memorize:

• Rectangle: P = 2(l + w), A = lw

• Square: P = 4s, A = s²

• Triangle: A = ½bh

• Circle: A = πr², C = 2πr

• Rectangular Prism: V = lwh

• Cube: V = s³

Solution: Exercises 4 to 5
4 Composite Shape Word Problem
Exercise 4
A picture frame is made of a rectangular piece of wood with dimensions 12 inches by 8 inches. A rectangular opening of 10 inches by 6 inches is cut out for the picture. What is the area of the wooden frame?
Visual: 🖼️ Frame with outer rectangle 12×8 and inner 10×6
Definition:

Composite Shape: A shape made of multiple basic shapes

Strategy: Find area of whole shape minus area of cut-out

Outer Area
12 × 8 = 96
Inner Area
10 × 6 = 60
Frame Area
96 - 60 = 36
Step 1: Calculate outer rectangle area

Outer area = 12 × 8 = 96 square inches

Step 2: Calculate inner rectangle area

Inner area = 10 × 6 = 60 square inches

Step 3: Find the frame area

Frame area = Outer area - Inner area

Frame area = 96 - 60 = 36 square inches

Frame Area = 36 in²
Final answer:

The area of the wooden frame is 36 square inches.

Applied rules:

• Composite Area: Total area - cut-out area

• Rectangle Area: A = l × w

• Subtraction: Finding remaining area

5 Real-World Application Problem
Exercise 5
A cylindrical water tank has a radius of 3 feet and a height of 8 feet. How many gallons of water can it hold? (1 cubic foot ≈ 7.48 gallons)
Visual: 💧 Cylindrical tank with r=3ft, h=8ft
Definition:

Cylinder Volume: V = πr²h

Unit Conversion: Converting between measurement systems

Volume in ft³
V = π × 3² × 8 ≈ 226.19
Convert to gallons
226.19 × 7.48 ≈ 1691.9
Step 1: Calculate cylinder volume

V = πr²h = π × 3² × 8

V = π × 9 × 8 = 72π ≈ 226.19 cubic feet

Step 2: Convert to gallons

Gallons = Cubic feet × conversion factor

Gallons = 226.19 × 7.48 ≈ 1,691.9 gallons

Step 3: Round appropriately

Since we're dealing with gallons, round to nearest whole number

Approximately 1,692 gallons

Tank Capacity ≈ 1,692 gallons
Final answer:

The tank can hold approximately 1,692 gallons of water.

Applied rules:

• Cylinder Volume: V = πr²h

• Unit Conversion: Multiply by conversion factor

• Real-World Context: Round appropriately

Comprehensive Guide: Geometry Word Problems
V = l × w × h
Volume Formula
Key definitions:

Word Problem: A mathematical problem presented in a narrative format that requires translation into mathematical operations

Perimeter: The total distance around the boundary of a 2D shape

Area: The measure of the surface enclosed by a 2D shape

Volume: The measure of space occupied by a 3D shape

Composite Shape: A shape formed by combining two or more basic geometric shapes

5-Step Problem-Solving Strategy:
  1. READ CAREFULLY: Understand what the problem is asking
  2. IDENTIFY KEY INFORMATION: List known facts and what needs to be found
  3. DRAW DIAGRAMS: Visualize the problem with sketches
  4. CHOOSE FORMULAS: Select appropriate geometric formulas
  5. SOLVE AND CHECK: Calculate and verify your answer makes sense
Tip 1: Look for key words like "perimeter", "area", "volume", "total", "difference".
Tip 2: Always write down units and keep them consistent throughout.
Tip 3: For composite shapes, break them into simpler shapes.
Tip 4: Estimate your answer before calculating to check reasonableness.
Common errors: Using wrong formulas, mixing units, misinterpreting the problem, calculation mistakes.
Success strategies: Practice regularly, draw diagrams, check work systematically.
Essential geometric formulas:

• Rectangle: P = 2(l + w), A = l × w

• Square: P = 4s, A = s²

• Triangle: A = ½bh

• Circle: A = πr², C = 2πr

• Rectangular Prism: V = l × w × h

• Cylinder: V = πr²h

• Composite shapes: Combine or subtract areas/volumes

Geometry Problem Comparison Chart
Exercise 6: Shape Comparisons
Compare different geometric calculations with same perimeter (20 units):
Square (side = 5), Rectangle (2×8), Equilateral Triangle (side ≈ 6.67)

Analysis: Shapes with the same perimeter can have different areas, demonstrating the relationship between perimeter and area.

  • Square: Area = 25 units²
  • Rectangle: Area = 16 units²
  • Triangle: Area ≈ 19.2 units²

Questions & Answers

Question: I often get confused about whether to use perimeter, area, or volume in word problems. How can I tell which one to use?

Answer: Look for specific keywords and context clues in the problem:

  • Perimeter: Words like "fence", "border", "around", "distance around", "trim", "edge"
  • Area: Words like "cover", "floor", "paint", "carpet", "space inside", "surface", "sod"
  • Volume: Words like "fill", "container", "capacity", "water", "cubic", "space inside 3D", "storage"

Also consider the dimensionality:

  • Perimeter: 1-dimensional measurement (linear units)
  • Area: 2-dimensional measurement (square units)
  • Volume: 3-dimensional measurement (cubic units)

Example: "Fencing a yard" → perimeter, "Carpeting a room" → area, "Filling a pool" → volume

Question: When I encounter composite shapes, how do I decide whether to add or subtract areas?

Answer: The decision depends on the specific problem context:

  • Add areas: When you want the total area of multiple separate shapes combined
  • Subtract areas: When one shape is cut out from another (like a hole, window, or frame)

Ask yourself: "Am I finding the area of everything together?" (add) or "Am I finding what remains after removing part of a shape?" (subtract)

Examples:

  • L-shaped room: Add the areas of the rectangles forming the L
  • Picture frame: Subtract the inner rectangle from the outer rectangle
  • Pool with a diving board platform: Add the pool area to the platform area

Drawing a diagram often clarifies whether to add or subtract.

Question: I struggle with multi-step word problems. Any advice for organizing my approach?

Answer: Use a systematic approach for multi-step problems:

  1. Read thoroughly: Read the problem at least twice
  2. List given information: Write down all known values and units
  3. Identify the goal: What exactly are you trying to find?
  4. Plan the steps: Determine the sequence of calculations needed
  5. Solve step-by-step: Work through each calculation carefully
  6. Check your work: Verify each step and the final answer

Example organization:

  • Given: [list all known information]
  • Find: [what the problem asks for]
  • Steps: [numbered list of calculations]
  • Check: [verify your answer makes sense]

This approach keeps you organized and reduces careless errors.