Perimeter of a Rectangle Answer Key - Solved Exercises

Complete answer key with 10 solved exercises on finding the perimeter of rectangles: basic problems, missing dimensions, word problems with detailed solutions.

Answer Key: Exercises 1 to 5
1 Basic Rectangle Perimeter
Exercise 1
Find the perimeter of a rectangle with length 8 cm and width 5 cm.
Definition:

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

Problem Context:

Perimeter measures the distance around the rectangle.

Perimeter calculation method:
  1. Identify the length and width of the rectangle
  2. Apply the perimeter formula: P = 2(l + w)
  3. Substitute the values: P = 2(8 + 5)
  4. Calculate: P = 2 × 13 = 26 cm
8×5
Given
l=8, w=5
Formula
P=2(l+w)
Calculate
P=26 cm
Step 1: Identify the dimensions: length = 8 cm, width = 5 cm
Step 2: Apply the perimeter formula: P = 2 × (8 + 5)
Step 3: Calculate: P = 2 × 13 = 26 cm
Perimeter = 26 cm
Final answer:

The perimeter of the rectangle is 26 cm.

Applied rules:

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

• Order of operations: Parentheses first

• Multiplication: Multiply by 2 after addition

2 Rectangle Perimeter with Different Units
Exercise 2
Find the perimeter of a rectangle with length 12 meters and width 7 meters.
Definition:

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

Problem Context:

Perimeter measures the distance around the rectangle in meters.

12×7
Given
l=12, w=7
Formula
P=2(l+w)
Calculate
P=38 m
Step 1: Identify the dimensions: length = 12 m, width = 7 m
Step 2: Apply the perimeter formula: P = 2 × (12 + 7)
Step 3: Calculate: P = 2 × 19 = 38 meters
Perimeter = 38 meters
Final answer:

The perimeter of the rectangle is 38 meters.

Applied rules:

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

• Unit consistency: All measurements in meters

• Final units: Answer in meters

3 Missing Length Problem
Exercise 3
A rectangle has a perimeter of 30 cm and a width of 6 cm. Find the length.
Definition:

Rectangle perimeter formula: P = 2(l + w), so l = (P/2) - w

Problem Context:

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

?×6
Given
P=30, w=6
Formula
l=(P/2)-w
Calculate
l=9 cm
Step 1: Write the perimeter formula: P = 2(l + w)
Step 2: Substitute known values: 30 = 2(l + 6)
Step 3: Solve for l: 30/2 = l + 6 → 15 = l + 6 → l = 9 cm
Length = 9 cm
Final answer:

The length of the rectangle is 9 cm.

Applied rules:

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

• Algebraic manipulation: Isolate unknown variable

• Division: Divide perimeter by 2

4 Missing Width Problem
Exercise 4
A rectangle has a perimeter of 42 cm and a length of 13 cm. Find the width.
Definition:

Rectangle perimeter formula: P = 2(l + w), so w = (P/2) - l

Problem Context:

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

13×?
Given
P=42, l=13
Formula
w=(P/2)-l
Calculate
w=8 cm
Step 1: Write the perimeter formula: P = 2(l + w)
Step 2: Substitute known values: 42 = 2(13 + w)
Step 3: Solve for w: 42/2 = 13 + w → 21 = 13 + w → w = 8 cm
Width = 8 cm
Final answer:

The width of the rectangle is 8 cm.

Applied rules:

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

• Algebraic manipulation: Isolate unknown variable

• Division: Divide perimeter by 2

5 Square Perimeter Problem
Exercise 5
Find the perimeter of a square with side length 9 cm.
Definition:

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

Problem Context:

A square is a special rectangle where all sides are equal.

9×9
Given
s=9
Formula
P=4s
Calculate
P=36 cm
Step 1: Identify the side length: all sides = 9 cm
Step 2: Apply the square perimeter formula: P = 4 × 9
Step 3: Calculate: P = 36 cm
Perimeter = 36 cm
Final answer:

The perimeter of the square is 36 cm.

Applied rules:

• Square perimeter formula: P = 4s

• Square property: All sides equal

• Multiplication: Multiply side by 4

Answer Key: Exercises 6 to 10
6 Decimal Rectangle Perimeter
Exercise 6
Find the perimeter of a rectangle with length 11.5 cm and width 8.7 cm.
Definition:

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

Problem Context:

Same formula applies, but with decimal measurements.

11.5×8.7
Given
l=11.5, w=8.7
Formula
P=2(l+w)
Calculate
P=40.4 cm
Step 1: Identify the dimensions: length = 11.5 cm, width = 8.7 cm
Step 2: Apply the perimeter formula: P = 2 × (11.5 + 8.7)
Step 3: Calculate: P = 2 × 20.2 = 40.4 cm
Perimeter = 40.4 cm
Final answer:

The perimeter of the rectangle is 40.4 cm.

Applied rules:

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

• Decimal addition: Align decimal points

• Decimal multiplication: Maintain decimal places

7 Fencing Problem
Exercise 7
A rectangular garden is 15 meters long and 10 meters wide. How much fencing is needed to enclose the garden?
Definition:

Fencing problem: Fencing needed equals the perimeter of the rectangle

Problem Context:

Perimeter represents the distance around the garden.

15×10
Given
l=15, w=10
Formula
P=2(l+w)
Fencing
P=50 m
Step 1: Identify the dimensions: length = 15 m, width = 10 m
Step 2: Apply the perimeter formula: P = 2 × (15 + 10)
Step 3: Calculate: P = 2 × 25 = 50 meters
Fencing needed = 50 meters
Final answer:

50 meters of fencing is needed to enclose the garden.

Applied rules:

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

• Real-world application: Perimeter represents boundary

• Unit consistency: Answer in meters

8 Picture Frame Perimeter
Exercise 8
A rectangular picture frame is 18 inches long and 12 inches wide. What is the perimeter of the frame?
Definition:

Frame perimeter: P = 2 × (length + width)

Problem Context:

Perimeter of the frame is the distance around the outside.

18×12
Given
l=18, w=12
Formula
P=2(l+w)
Calculate
P=60 in
Step 1: Identify the dimensions: length = 18 in, width = 12 in
Step 2: Apply the perimeter formula: P = 2 × (18 + 12)
Step 3: Calculate: P = 2 × 30 = 60 inches
Perimeter = 60 inches
Final answer:

The perimeter of the picture frame is 60 inches.

Applied rules:

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

• Real-world context: Frame goes around the picture

• Unit consistency: Answer in inches

9 Rectangle Area Given Perimeter
Exercise 9
A rectangle has a perimeter of 36 cm and a length of 10 cm. Find both the width and the area of the rectangle.
Definition:

Combined problem: First find width using P = 2(l + w), then find area using A = lw

Problem Context:

Multi-step problem requiring perimeter and area formulas.

10×?
Given
P=36, l=10
Find width
w=8 cm
Find area
A=80 cm²
Step 1: Find width using perimeter formula: 36 = 2(10 + w) → w = 8 cm
Step 2: Find area using area formula: A = 10 × 8 = 80 cm²
Step 3: Answer: width = 8 cm, area = 80 cm²
Width = 8 cm, Area = 80 cm²
Final answer:

The width of the rectangle is 8 cm and the area is 80 cm².

Applied rules:

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

• Rectangle area formula: A = lw

• Multi-step problem: Solve sequentially

10 Perimeter Comparison Problem
Exercise 10
Rectangle A has length 12 cm and width 8 cm. Rectangle B has length 10 cm and width 10 cm. Which rectangle has a greater perimeter?
Definition:

Comparison problem: Calculate both perimeters and compare

Problem Context:

Compare perimeters of two different rectangles.

12×8
10×10
Rectangle A
P=40 cm
Rectangle B
P=40 cm
Compare
Equal
Step 1: Calculate Rectangle A perimeter: P = 2(12 + 8) = 40 cm
Step 2: Calculate Rectangle B perimeter: P = 2(10 + 10) = 40 cm
Step 3: Compare: Both perimeters equal 40 cm
Both perimeters equal 40 cm
Final answer:

Both rectangles have the same perimeter of 40 cm.

Applied rules:

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

• Comparison: Calculate each separately

• Equality: Same perimeter despite different dimensions

Rectangle Perimeter Formulas and Properties
P = 2(l + w)
Rectangle Perimeter Formula
Standard Form
P = 2(l + w)
Perimeter equals twice the sum of length and width
Expanded Form
P = 2l + 2w
Perimeter equals twice the length plus twice the width
Square Form
P = 4s
Perimeter of square equals four times the side length
l = (P/2) - w
Finding Missing Length
From Perimeter
l = (P/2) - w
Length equals half perimeter minus width
From Perimeter
w = (P/2) - l
Width equals half perimeter minus length
Opposite sides
Equal
Angles
Right (90°)
Diagonals
Equal
Key concept: Perimeter measures distance around the rectangle.
Unit: Always express perimeter in linear units (cm, m, in, ft).
Formula: P = 2(l + w) works for all rectangles.
Essential Rectangle Perimeter Rules:

• Basic rule: P = 2(l + w)

• Alternative: P = 2l + 2w

• Square rule: P = 4s

• Missing dimension: l = (P/2) - w

• Missing dimension: w = (P/2) - l

Detailed Summary: Rectangle Perimeter Answer Key
P = 2(l + w)
Core Rectangle Formula
Key definitions:

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

Rectangle: A four-sided shape with four right angles and opposite sides equal

Length: The longer side of a rectangle

Width: The shorter side of a rectangle

Formula: A mathematical rule expressed using symbols

Perimeter of rectangle: The sum of all four sides (2l + 2w)

Rectangle perimeter calculation methodology:
  1. Identify the rectangle: Confirm it has four right angles and opposite sides equal
  2. Measure or identify dimensions: Get the length and width measurements
  3. Apply the correct formula: Use P = 2(l + w) for rectangles
  4. Substitute values: Replace l and w with actual measurements
  5. Calculate: Follow order of operations (parentheses first)
  6. Check units: Ensure answer has correct linear units
  7. Verify answer: Check if answer makes sense in context
Tip 1: Always identify length and width before applying the formula.
Tip 2: Remember that opposite sides of a rectangle are equal.
Tip 3: Check that all measurements are in the same unit before calculating.
Tip 4: For squares, remember that length = width, so P = 4s.
Tip 5: When finding missing dimensions, use algebra to isolate the variable.
Tip 6: Always include units in your final answer.
Common errors: Confusing length and width, forgetting to multiply by 2, mixing units, calculation mistakes.
Problem-solving strategy: Draw the rectangle, label dimensions, apply formula, check answer.
Rectangle Perimeter Formulas:

• Standard formula: P = 2(l + w)

• Expanded formula: P = 2l + 2w

• Square formula: P = 4s

• Find length: l = (P/2) - w

• Find width: w = (P/2) - l

Simple Examples

Rectangle: l=5, w=3 → P = 2(5+3) = 16

Square: s=4 → P = 4×4 = 16

Rectangle: l=7, w=2 → P = 2(7+2) = 18

Advanced Examples

Find width: P=24, l=8 → w = (24/2)-8 = 4

Find length: P=30, w=6 → l = (30/2)-6 = 9

Decimal rectangle: l=5.5, w=3.2 → P = 2(5.5+3.2) = 17.4

Answer Key Summary:
Ex 1
26 cm
Ex 2
38 m
Ex 3
9 cm
Ex 4
8 cm
Ex 5
36 cm
Ex 6
40.4 cm
Ex 7
50 m
Ex 8
60 in
Ex 9
w=8, A=80
Ex 10
40 cm each

Questions & Answers

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

Answer: Great question! We multiply by 2 because a rectangle has four sides: two lengths and two widths.

  • A rectangle has four sides: top, bottom, left, and right
  • The top and bottom sides are equal (both are the length)
  • The left and right sides are equal (both are the width)
  • So the perimeter is: length + length + width + width = 2l + 2w = 2(l + w)

If you only added length + width once, you'd only be counting two sides, not all four sides!

Question: My child is having trouble with missing dimension problems. How can I help them understand finding length or width when given perimeter?

Answer: Here are effective strategies to help your child understand missing dimension problems:

  • Start with the formula: P = 2(l + w), so if you know P and w, you can find l
  • Use substitution: Replace known values in the formula
  • Solve step by step: First divide perimeter by 2, then subtract the known dimension
  • Check work: Once you find the missing dimension, substitute back to verify
  • Practice with concrete examples: Use real rectangles and measure them

The key is understanding that the perimeter formula can be rearranged to solve for any missing dimension!

Question: I sometimes forget which is length and which is width. How can I remember?

Answer: Here are memory tricks to remember length and width:

  • Length is LONGER: The length is usually the longer side of the rectangle
  • Width is WIDER: The width is how wide the rectangle is (usually shorter)
  • Think of a door: It's taller (length) than it is thick (width)
  • Think of a table: It's longer than it is wide

Remember: it doesn't matter which is which in the formula P = 2(l + w) because addition is commutative! As long as you have both measurements, you'll get the right answer.