Solved Exercises on Finding Perimeter Geometry in Grade 3

Master finding perimeter geometry: rectangles, squares, polygons, real-life applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle perimeter
Exercise 1
Find the perimeter of a rectangle with length 7 units and width 5 units.
Definition:

Rectangle: A four-sided shape with opposite sides equal and all angles 90°

Perimeter: The total distance around the rectangle

Length: The longer side of the rectangle

Width: The shorter side of the rectangle

Perimeter method:
  1. Add up all four sides: length + width + length + width
  2. Or use formula: Perimeter = 2 × (length + width)
Given
Length = 7, Width = 5
Formula
P = 2(l+w)
Result
P = 24
Step 1: Identify the dimensions

Length = 7 units, Width = 5 units

Step 2: Apply the perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (7 + 5)

Step 3: Calculate the result

Perimeter = 2 × 12 = 24 units

Perimeter = 24 units
Final answer:

The perimeter of the rectangle is 24 units

Applied rules:

• Rectangle Property: Opposite sides are equal

• Perimeter Formula: P = 2(l+w) where l is length and w is width

• Order of Operations: Parentheses first, then multiplication

2 Square perimeter
Exercise 2
Find the perimeter of a square with side length 6 units.
Definition:

Square: A special rectangle where all four sides are equal

Side: Any side of the square (all are equal)

Given
Side = 6 units
Formula
P = 4s
Result
P = 24
Step 1: Identify the dimensions

Since it's a square, all sides = 6 units

Length = 6 units, Width = 6 units

Step 2: Apply the perimeter formula

Method 1: P = 2(l + w) = 2(6 + 6) = 2(12) = 24

Method 2: P = 4s = 4(6) = 24 (since all sides are equal)

Step 3: Calculate the result

Perimeter = 24 units

Perimeter = 24 units
Final answer:

The perimeter of the square is 24 units

Applied rules:

• Square Property: All four sides are equal

• Square Perimeter: P = 4s where s is the side length

• Rectangle Perimeter: P = 2(l+w) also works for squares

3 Triangle perimeter
Exercise 3
Find the perimeter of a triangle with sides measuring 3 units, 4 units, and 5 units.
Definition:

Triangle: A three-sided polygon with three angles

Perimeter of Polygon: The sum of all side lengths

Given
Sides = 3, 4, 5 units
Formula
P = a+b+c
Result
P = 12
Step 1: Identify all side lengths

Side 1 = 3 units, Side 2 = 4 units, Side 3 = 5 units

Step 2: Apply the perimeter formula for polygons

Perimeter = sum of all side lengths

Perimeter = 3 + 4 + 5

Step 3: Calculate the result

Perimeter = 12 units

Step 4: Note special case

This is a right triangle since 3² + 4² = 9 + 16 = 25 = 5²

Perimeter = 12 units
Final answer:

The perimeter of the triangle is 12 units

Applied rules:

• Polygon Perimeter: Sum of all side lengths

• Triangle Property: Three-sided polygon

• Pythagorean Triple: 3-4-5 is a right triangle

Rules and methods, laws,...
\(P = 2(l + w), P = 4s, P = a + b + c\)
Perimeter Formulas
Rectangle
\(P = 2(l + w)\)
Opposite sides equal
Square
\(P = 4s\)
All sides equal
Triangle
\(P = a + b + c\)
Three sides
Polygon
\(P = \text{sum of sides}\)
Any number of sides
Key definitions:

Perimeter: The total distance around the outside of a two-dimensional shape

Geometry: The branch of mathematics dealing with shapes, sizes, and properties of figures

Rectangle: A four-sided shape with opposite sides equal and all angles 90°

Square: A rectangle with all sides equal

Triangle: A three-sided polygon

Polygon: A closed shape with straight sides

Complete methodology:
  1. Identify the shape: Determine if it's a rectangle, square, triangle, or other polygon
  2. Count the sides: Count how many sides the shape has
  3. Measure dimensions: Find the length of each side
  4. Select formula: Use the appropriate perimeter formula
  5. Substitute values: Put the measurements into the formula
  6. Calculate: Perform the arithmetic
  7. Verify: Check by adding all sides individually
Tip 1: Remember that perimeter is always a linear measurement.
Tip 2: For any polygon, perimeter is the sum of all side lengths.
Tip 3: A square is a special type of rectangle where all sides are equal.
Tip 4: Don't forget to include the unit of measurement in your answer.
Common errors: Forgetting to multiply by 2 for rectangles, confusing perimeter with area, miscounting sides.
Memory aids: "Perimeter is the fence around the yard" and "add up all the sides".
Formulas to know by heart:

• Perimeter of rectangle: P = 2(l + w)

• Perimeter of square: P = 4s

• Perimeter of triangle: P = a + b + c

• Perimeter of any polygon: Sum of all sides

Solution: Exercises 4 to 5
4 Real-world application
Exercise 4
A rectangular garden has a length of 12 feet and a width of 8 feet. How much fencing is needed to go around the garden?
Definition:

Word problem: A math problem presented in a real-world context

Fencing: Material needed to surround the garden, which equals the perimeter

Given
Length = 12 ft, Width = 8 ft
Formula
P = 2(l+w)
Result
P = 40 ft
Step 1: Identify what's being asked

The problem asks for the amount of fencing needed to go around the garden

This means we need to find the perimeter of the rectangular garden

Step 2: Identify the dimensions

Garden length = 12 feet, Garden width = 8 feet

Step 3: Apply the perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (12 + 8)

Step 4: Calculate the result

Perimeter = 2 × 20 = 40 feet

Step 5: State the answer in context

40 feet of fencing is needed to go around the garden

Fencing needed = 40 feet
Final answer:

40 feet of fencing is needed to go around the garden

Applied rules:

• Real-world modeling: Representing physical situations with geometric concepts

• Perimeter application: Perimeter represents the distance around a shape

• Unit awareness: Including measurement units in the final answer

5 Comparison problem
Exercise 5
Rectangle A has length 9 cm and width 5 cm. Square B has sides of 7 cm. Which shape has a larger perimeter?
Definition:

Comparison problem: Calculating and comparing measurements of different shapes

Rectangle A: Standard rectangle with different length and width

Square B: All sides equal

Rectangle A
P = 2(9+5) = 28cm
Square B
P = 4×7 = 28cm
Result
Equal perimeters
Step 1: Calculate Rectangle A's perimeter

Rectangle A: length = 9 cm, width = 5 cm

Perimeter = 2 × (length + width) = 2 × (9 + 5) = 2 × 14 = 28 cm

Step 2: Calculate Square B's perimeter

Square B: side length = 7 cm

Perimeter = 4 × side = 4 × 7 = 28 cm

Step 3: Compare the perimeters

Rectangle A: 28 cm

Square B: 28 cm

28 cm = 28 cm

Step 4: State the conclusion

Both shapes have the same perimeter of 28 cm

Both shapes have equal perimeters (28 cm each)
Final answer:

Both shapes have the same perimeter of 28 cm

Applied rules:

• Comparison strategy: Calculate both values before comparing

• Shape identification: Recognizing different geometric shapes

• Units consistency: Both measurements in the same unit

Complete Guide: Laws, Methods, Rules, Definitions, Expressions
\(P = 2(l + w), P = 4s, P = \text{sum of sides}\)
Perimeter Formulas for Various Shapes
Key definitions:

Perimeter: The total distance around the outside edge of a two-dimensional shape, measured in linear units (inches, feet, centimeters, etc.).

Geometry: The branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.

Rectangle: A quadrilateral with four right angles (90°) and opposite sides equal in length.

Square: A rectangle with all four sides equal in length.

Triangle: A polygon with three sides and three angles.

Polygon: A closed plane figure bounded by straight sides.

Complete methodology:
  1. Identify the shape: Determine the type of geometric figure
  2. Count the sides: Determine how many sides the shape has
  3. Measure or identify dimensions: Find the length of each side
  4. Select the appropriate formula: Use the correct perimeter formula for the shape
  5. Substitute values: Replace variables with actual measurements
  6. Perform calculations: Add or multiply as required by the formula
  7. State the answer: Include the appropriate unit of measurement
  8. Verify the result: Optionally add all sides individually to confirm
Tip 1: Remember that for any polygon, perimeter is always the sum of all side lengths.
Tip 2: A square is a special rectangle where all four sides are equal. You can use either P = 2(l + w) or P = 4s for squares.
Tip 3: Always include the unit of measurement (inches, feet, centimeters, etc.) in your final answer.
Tip 4: You can check your answer by adding all sides individually: length + width + length + width for rectangles.
Rectangle Properties: 4 right angles, opposite sides equal, diagonals equal and bisect each other, sum of interior angles is 360°.
Square Properties: 4 equal sides, 4 right angles, diagonals equal and perpendicular, special type of rectangle and rhombus.
Triangle Properties: 3 sides, 3 angles, sum of interior angles is 180°, classified by sides or angles.
Common Errors: Confusing perimeter with area, forgetting to multiply by 2 in rectangle formula, misidentifying shape type.
Essential formulas:

• Perimeter of rectangle: P = 2(l + w)

• Perimeter of square: P = 4s

• Perimeter of triangle: P = a + b + c

• Perimeter of any polygon: Sum of all sides

• Area of rectangle: A = l × w

• Area of square: A = s²

Perimeter Geometry Infographic
📐
Shape Types

• Rectangle: 2 lengths + 2 widths

• Square: 4 equal sides

• Triangle: 3 sides

• Polygon: Any number of sides

Perimeter Formulas

• Rectangle: P = 2(l + w)

• Square: P = 4s

• Triangle: P = a + b + c

• Any polygon: Sum all sides

Verification

• Add all sides: l + w + l + w

• Should equal 2(l + w)

• Check arithmetic carefully

• Include units in answer

Problem-Solving Workflow
1
Identify Shape
2
Count Sides
3
Get Dimensions
4
Apply Formula
5
Calculate & Check
Visual Representation

Various shapes showing perimeter as the boundary

Rectangle: P = 2(l + w)

Square: P = 4s

Triangle: P = a + b + c

Remember: Perimeter is always the distance around the shape!

Questions & Answers

Question: What's the difference between perimeter and area? I always get them mixed up.

Answer: This is a very common question! Here's the key difference:

  • Perimeter: The distance around the outside of a shape. Think of it as walking around the edge of a garden. It's measured in linear units (inches, feet, centimeters).
  • Area: The amount of space inside a shape. Think of it as covering the floor with tiles. It's measured in square units (square inches, square feet, square centimeters).

For a rectangle: Perimeter = 2(length + width) [linear measurement] Area = length × width [square measurement]

A helpful memory trick: "Perimeter" starts with "P" like "Periscope" - you're looking around the edge. "Area" sounds like "are" - "How much area is covered?"

Question: Why do we multiply by 2 in the rectangle perimeter formula? Can't we just add the length and width once?

Answer: Great question! We multiply by 2 because a rectangle has TWO lengths and TWO widths. Let's break it down:

  • A rectangle has 4 sides: top, bottom, left, right
  • The top and bottom sides are both equal to the length
  • The left and right sides are both equal to the width
  • So the total perimeter is: length + length + width + width = 2×length + 2×width = 2(length + width)

If you only added length + width, you'd only be counting half the perimeter! For example, if length = 4 and width = 3: Using P = l + w: 4 + 3 = 7 (incorrect!) Using P = 2(l + w): 2(4 + 3) = 2(7) = 14 (correct!)

You can verify: 4 + 3 + 4 + 3 = 14 ✓

Question: Is a square really a rectangle? I thought they were different shapes.

Answer: Yes, a square IS a special type of rectangle! Here's why:

  • A rectangle is defined as a four-sided shape with all angles equal to 90° (right angles)
  • A square meets this definition: it has four sides and all four angles are 90°
  • But a square has an extra property: all four sides are equal in length

Think of it this way: all squares are rectangles, but not all rectangles are squares. It's like how all dogs are animals, but not all animals are dogs.

This is why the rectangle perimeter formula P = 2(l + w) works for squares too. If all sides are equal (let's call that length 's'), then P = 2(s + s) = 2(2s) = 4s, which is the square perimeter formula!