Solved Exercises on Problem Solving with Plane Figures in Grade 3

Master problem solving: triangles, quadrilaterals, area, perimeter, and geometric relationships through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Perimeter Problem
Exercise 1
Sarah wants to put a fence around her rectangular garden. The garden is 8 meters long and 5 meters wide. How much fencing does she need?
Definition:

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

Solution method:
  1. Identify the shape (rectangle)
  2. Identify the dimensions (length = 8m, width = 5m)
  3. Apply the perimeter formula
  4. Calculate the result
Dimensions
Length = 8m, Width = 5m
Formula
P = 2(l+w)
Result
P = 26m
Step 1: Identify the problem

Sarah needs fencing around a rectangular garden, which means finding the perimeter.

Step 2: Identify the dimensions

Length = 8 meters, Width = 5 meters

Step 3: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width) = 2 × (8 + 5) = 2 × 13 = 26 meters

Step 4: State the answer

Sarah needs 26 meters of fencing.

Sarah needs 26 meters of fencing
Final answer:

Sarah needs 26 meters of fencing because the perimeter of her rectangular garden is 26 meters.

Applied rules:

• Rectangle perimeter: P = 2 × (length + width)

• Problem identification: Fencing = perimeter

• Unit consistency: Answer in meters

2 Shape Classification Problem
Exercise 2
Tom sees a shape with 4 equal sides and 4 right angles. What shape is it? What is the perimeter if each side is 6 cm?
Definition:

Square: A quadrilateral with 4 equal sides and 4 right angles. Perimeter of square: P = 4 × side length.

Properties
4 equal sides, 4 right angles
Shape
Square
Perimeter
4 × 6cm = 24cm
Step 1: Identify the shape

A shape with 4 equal sides and 4 right angles is a square.

Step 2: Apply the square perimeter formula

Perimeter = 4 × side length = 4 × 6 cm = 24 cm

Step 3: State the answers

The shape is a square with a perimeter of 24 cm.

Shape: Square, Perimeter: 24 cm
Final answer:

The shape is a square. Its perimeter is 24 cm because P = 4 × 6 cm = 24 cm.

Applied rules:

• Square definition: 4 equal sides and 4 right angles

• Square perimeter: P = 4 × side length

• Shape recognition: Match properties to definitions

3 Area Comparison Problem
Exercise 3
Lisa has two gardens: Garden A is a square with sides of 4 meters, and Garden B is a rectangle with length 6 meters and width 3 meters. Which garden has a larger area?
Definition:

Area: The amount of space inside a shape. Square area: A = side × side. Rectangle area: A = length × width.

Garden A
Square, 4m sides
Area A
4 × 4 = 16m²
Area B
6 × 3 = 18m²
Step 1: Calculate Garden A's area

Garden A is a square with sides of 4 meters

Area of square = side × side = 4m × 4m = 16m²

Step 2: Calculate Garden B's area

Garden B is a rectangle with length 6m and width 3m

Area of rectangle = length × width = 6m × 3m = 18m²

Step 3: Compare the areas

Garden A: 16m², Garden B: 18m²

Since 18m² > 16m², Garden B has a larger area.

Garden B has the larger area (18m²)
Final answer:

Garden B has the larger area. Garden A has 16m² while Garden B has 18m².

Applied rules:

• Square area formula: A = side²

• Rectangle area formula: A = length × width

• Comparison: Compare numerical results

Rules and methods, laws,...
P = 2(l+w), A = l×w
Perimeter and Area Formulas
Rectangle
P = 2(l+w), A = l×w
Perimeter and Area
Square
P = 4s, A = s²
Perimeter and Area
Triangle
P = a+b+c
Perimeter only
Key definitions:

Perimeter: The distance around the outside of a shape

Area: The amount of space inside a shape

Problem solving: Using math skills to answer real-world questions

Complete methodology:
  1. Read the problem carefully: Identify what is being asked
  2. Identify the shape(s): Recognize the geometric figure
  3. Find the dimensions: Locate length, width, sides
  4. Choose the formula: Select the correct mathematical operation
  5. Calculate: Perform the computation
  6. Check: Verify the answer makes sense
Tip 1: Draw a picture to visualize the problem.
Tip 2: Label dimensions clearly on your drawing.
Tip 3: Always include units in your final answer.
Tip 4: Check if your answer is reasonable for the situation.
Common errors: Mixing up perimeter and area formulas, forgetting units, miscalculating, misreading problems.
Exam preparation: Memorize formulas, practice word problems, check answers, use drawings.
Properties to know by heart:

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

• Rectangle area: A = length × width

• Square perimeter: P = 4 × side

• Square area: A = side × side

• Triangle perimeter: P = side1 + side2 + side3

Detailed Summary: Problem Solving with Plane Figures

Definitions

Plane figure: A flat shape that lies entirely in one plane.

Perimeter: The total distance around the edge of a shape.

Area: The amount of surface inside a shape, measured in square units.

Problem solving: Using mathematical knowledge to find solutions to real-world situations.

Dimensions: Measurements like length, width, and height that describe a shape.

Core Rules

• Perimeter = distance around shape

• Area = space inside shape

• Rectangle perimeter: P = 2(l + w)

• Rectangle area: A = l × w

• Square perimeter: P = 4s

• Square area: A = s²

Problem-Solving Steps

1. Read the problem carefully and underline key information

2. Identify what is being asked (perimeter? area? shape?)

3. Draw a diagram and label known dimensions

4. Choose the correct formula

5. Substitute values and calculate

6. Check if the answer makes sense

Examples

Simple problem: Square with side 5m → Perimeter = 4×5 = 20m

Word problem: Fencing around rectangle → Use perimeter formula

Comparison: Which garden is larger? → Calculate both areas

Real-world: Carpeting a room → Use area formula

Tips & Tricks

• Draw pictures for every problem

• Write down the formula before substituting numbers

• Always include units in your answer

• Estimate before calculating to check reasonableness

• Circle or highlight important numbers in word problems

Key Takeaways

• Understand the difference between perimeter and area

• Memorize the basic formulas

• Always read problems carefully

• Draw diagrams to visualize

• Check your work before finishing

Additional Learning Materials
A = l × w, P = 2(l+w)
Area and Perimeter Formulas
Advanced definitions:

Circumference: Perimeter of a circle

Surface area: Total area of all surfaces of a 3D shape

Volume: Amount of space inside a 3D shape

Advanced problem-solving strategies:
  1. Work backwards: Start with the answer and find the path
  2. Make a table: Organize information systematically
  3. Look for patterns: Identify repeated relationships
  4. Estimate first: Predict the approximate answer
  5. Check with different methods: Verify using alternative approaches
Advanced Tip 1: When comparing shapes, calculate both area and perimeter separately.
Advanced Tip 2: For complex shapes, break them into simpler shapes.
Advanced Tip 3: Always match units when comparing different measurements.
Advanced Tip 4: Practice converting between different units (cm to m, etc.).
Real-world connections: Architects, engineers, and designers use these concepts daily.
Advanced applications: Construction, landscaping, interior design, and manufacturing.
Additional formulas:

• Triangle perimeter: P = a + b + c

• Triangle area: A = ½ × base × height

• Perimeter of any polygon: Sum of all sides

• Area of composite shapes: Break into simpler shapes

• Unit conversion: 1m = 100cm, 1m² = 10,000cm²

Questions & Answers

Question: I get confused about when to use perimeter vs area. How do I know which one to calculate?

Answer: Here's a simple way to remember:

  • Perimeter: Use when you need to go "around" something - fencing, border, trim, walking around the edge
  • Area: Use when you need to cover "inside" something - carpeting, painting, planting grass

Memory tricks:

  • "Perimeter" starts with "P" like "Picture frame around" or "Periscope goes around"
  • "Area" sounds like "Are-A" - "What are the A-mount of space inside?"

Practice questions: If you want to put ribbon around a poster, use perimeter. If you want to paint the whole poster, use area.

Question: How can I solve word problems faster? They take me too long to figure out.

Answer: Here are strategies to solve word problems more efficiently:

  • Circle key numbers: Identify the important measurements
  • Underline the question: Know exactly what you need to find
  • Draw a quick sketch: Visualize the problem
  • Write the formula first: Don't substitute numbers until you have the right formula
  • Check for units: Make sure your answer has the right unit

Practice reading the question twice: once to understand, once to identify what's needed.

Start with simpler problems and gradually increase difficulty as you get faster.

Question: What should I do if the problem gives mixed units (like meters and centimeters)?

Answer: Always convert all measurements to the same unit before calculating! Here's the process:

  • Identify all units: Find which units are used in the problem
  • Choose your target unit: Decide which unit to convert everything to
  • Convert: Use conversion factors (1m = 100cm, 1km = 1000m)
  • Calculate: Now that all measurements are in the same unit

Common conversions:

  • 1 meter = 100 centimeters
  • 1 kilometer = 1000 meters
  • 1 foot = 12 inches

If the answer asks for a specific unit, make sure to provide it in that unit!

Question: How can I check if my answer is reasonable? Sometimes I make calculation mistakes.

Answer: Here are effective checking strategies:

  • Estimate first: Round numbers and do mental math to predict the answer
  • Check units: Does your answer have the correct unit?
  • Does it make sense?: Would a 1000-meter garden fence be reasonable for a small yard?
  • Reverse calculation: Work backwards to see if you get the original values
  • Alternative method: Solve using a different approach

For example, if you calculated a rectangle's area as 20m², estimate: 4m × 5m ≈ 20m² ✓

Always spend 30 seconds reviewing your work - it catches many common errors!

Question: Where do I see plane figure problems in real life? This helps me understand why they matter!

Answer: Plane figure problems are everywhere in daily life! Here are examples:

  • Home improvement: Calculating paint needed for walls, carpet for floors
  • Gardening: Fencing around gardens, soil for planters
  • Shopping: Comparing pizza sizes, rug dimensions
  • Construction: Building foundations, room layouts
  • Arts and crafts: Framing pictures, designing projects

Try this: Measure your bedroom and calculate how much carpet you'd need to cover the floor!

These skills help you make informed decisions about purchases and projects!