Perimeter and Area of a Rectangle - Solved Exercises

Complete practice set with 10 exercises on finding both perimeter and area of rectangles: basic problems, missing dimensions, word problems with detailed solutions and methods.

Solution: Exercises 1 to 5
1 Rectangle Perimeter and Area
Exercise 1
Find both the perimeter and area of a rectangle with length 6 cm and width 4 cm.
Definition:

Rectangle perimeter: P = 2 × (length + width) or P = 2l + 2w

Rectangle area: A = length × width or A = lw

Rectangle measurement method:
  1. Identify the length and width of the rectangle
  2. Apply the perimeter formula: P = 2(l + w)
  3. Apply the area formula: A = l × w
  4. Calculate both values using correct arithmetic
6×4
Given
l=6, w=4
Perimeter
P=20 cm
Area
A=24 cm²
Step 1: Identify the dimensions: length = 6 cm, width = 4 cm
Step 2: Calculate perimeter: P = 2 × (6 + 4) = 20 cm
Step 3: Calculate area: A = 6 × 4 = 24 cm²
Perimeter = 20 cm, Area = 24 cm²
Final answer:

The perimeter of the rectangle is 20 cm and the area is 24 cm².

Applied rules:

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

• Rectangle area formula: A = lw

• Unit consistency: Perimeter in cm, area in cm²

2 Rectangle with Different Units
Exercise 2
Find the perimeter and area of a rectangle with length 9 meters and width 5 meters.
Definition:

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

Rectangle area formula: A = length × width

9×5
Given
l=9, w=5
Perimeter
P=28 m
Area
A=45 m²
Step 1: Identify the dimensions: length = 9 m, width = 5 m
Step 2: Apply perimeter formula: P = 2 × (9 + 5)
Step 3: Apply area formula: A = 9 × 5
Step 4: Calculate: P = 28 m, A = 45 m²
Perimeter = 28 m, Area = 45 m²
Final answer:

The perimeter of the rectangle is 28 meters and the area is 45 square meters.

Applied rules:

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

• Rectangle area formula: A = lw

• Unit consistency: All measurements in meters

3 Missing Length Problem
Exercise 3
A rectangle has a perimeter of 28 cm and a width of 6 cm. Find the length and the area.
Definition:

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

Rectangle area formula: A = l × w

?×6
Given
P=28, w=6
Find length
l=8 cm
Find area
A=48 cm²
Step 1: Find the length using perimeter formula: 28 = 2(l + 6)
Step 2: Solve for l: 28/2 = l + 6 → 14 = l + 6 → l = 8 cm
Step 3: Calculate area: A = 8 × 6 = 48 cm²
Length = 8 cm, Area = 48 cm²
Final answer:

The length of the rectangle is 8 cm and the area is 48 cm².

Applied rules:

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

• Algebraic manipulation: Isolate unknown variable

• Rectangle area formula: A = lw

4 Missing Width and Area
Exercise 4
A rectangle has a perimeter of 34 cm and a length of 10 cm. Find the width and the area.
Definition:

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

Rectangle area formula: A = l × w

10×?
Given
P=34, l=10
Find width
w=7 cm
Find area
A=70 cm²
Step 1: Find width using perimeter formula: 34 = 2(10 + w)
Step 2: Solve for w: 34/2 = 10 + w → 17 = 10 + w → w = 7 cm
Step 3: Calculate area: A = 10 × 7 = 70 cm²
Width = 7 cm, Area = 70 cm²
Final answer:

The width of the rectangle is 7 cm and the area is 70 cm².

Applied rules:

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

• Algebraic manipulation: Isolate unknown variable

• Rectangle area formula: A = lw

5 Square Perimeter and Area
Exercise 5
Find the perimeter and area of a square with side length 7 cm.
Definition:

Square perimeter formula: P = 4 × side length or P = 4s

Square area formula: A = side × side or A = s²

7×7
Given
s=7
Perimeter
P=28 cm
Area
A=49 cm²
Step 1: Identify the side length: All sides equal 7 cm
Step 2: Apply square perimeter formula: P = 4 × 7
Step 3: Apply square area formula: A = 7²
Step 4: Calculate: P = 28 cm, A = 49 cm²
Perimeter = 28 cm, Area = 49 cm²
Final answer:

The perimeter of the square is 28 cm and the area is 49 cm².

Applied rules:

• Square perimeter formula: P = 4s

• Square area formula: A = s²

• Square property: All sides equal

Solution: Exercises 6 to 10
6 Decimal Rectangle Calculations
Exercise 6
Find the perimeter and area of a rectangle with length 8.5 cm and width 6.3 cm.
Definition:

Rectangle with decimals: Same formulas apply to decimal measurements

8.5×6.3
Given
l=8.5, w=6.3
Perimeter
P=29.6 cm
Area
A=53.55 cm²
Step 1: Identify the dimensions: length = 8.5 cm, width = 6.3 cm
Step 2: Calculate perimeter: P = 2 × (8.5 + 6.3) = 29.6 cm
Step 3: Calculate area: A = 8.5 × 6.3 = 53.55 cm²
Perimeter = 29.6 cm, Area = 53.55 cm²
Final answer:

The perimeter of the rectangle is 29.6 cm and the area is 53.55 cm².

Applied rules:

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

• Rectangle area formula: A = lw

• Decimal arithmetic: Align decimal points correctly

7 Fencing and Flooring Problem
Exercise 7
A rectangular garden is 15 meters long and 10 meters wide. How much fencing is needed for the perimeter and how much area is available for planting?
Definition:

Real-world application: Fencing = perimeter, planting area = rectangle area

15×10
Given
l=15, w=10
Fencing
P=50 m
Planting
A=150 m²
Step 1: Identify the dimensions: length = 15 m, width = 10 m
Step 2: Calculate perimeter for fencing: P = 2 × (15 + 10) = 50 m
Step 3: Calculate area for planting: A = 15 × 10 = 150 m²
Fencing needed = 50 m, Planting area = 150 m²
Final answer:

50 meters of fencing is needed and 150 square meters are available for planting.

Applied rules:

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

• Rectangle area formula: A = lw

• Real-world context: Perimeter = boundary, Area = interior space

8 Picture Frame Dimensions
Exercise 8
A rectangular picture frame is 16 inches long and 11 inches wide. What is the perimeter of the frame and the area of the picture inside?
Definition:

Frame perimeter: Distance around the outside of the frame

Picture area: Space inside the frame

16×11
Given
l=16, w=11
Frame perimeter
P=54 in
Picture area
A=176 in²
Step 1: Identify the dimensions: length = 16 in, width = 11 in
Step 2: Calculate frame perimeter: P = 2 × (16 + 11) = 54 in
Step 3: Calculate picture area: A = 16 × 11 = 176 in²
Frame perimeter = 54 in, Picture area = 176 in²
Final answer:

The perimeter of the frame is 54 inches and the area of the picture is 176 square inches.

Applied rules:

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

• Rectangle area formula: A = lw

• Real-world context: Frame boundary vs. interior space

9 Rectangle with Known Area and Perimeter
Exercise 9
A rectangle has a perimeter of 32 cm and an area of 60 cm². Find the length and width of the rectangle.
Definition:

System of equations: Use both P = 2(l + w) and A = lw to solve for dimensions

?×?
Given
P=32, A=60
System
l+w=16, lw=60
Solution
l=10, w=6
Step 1: From perimeter: 32 = 2(l + w) → l + w = 16
Step 2: From area: 60 = l × w
Step 3: Solve system: l = 16 - w, substitute into area equation: (16 - w)w = 60 → w² - 16w + 60 = 0 → w = 6, l = 10
Length = 10 cm, Width = 6 cm
Final answer:

The rectangle has length 10 cm and width 6 cm.

Applied rules:

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

• Rectangle area formula: A = lw

• System of equations: Solve using substitution method

10 Perimeter and Area Comparison
Exercise 10
Rectangle A has dimensions 12 cm × 8 cm. Rectangle B has dimensions 10 cm × 10 cm. Compare their perimeters and areas.
Definition:

Comparison problem: Calculate both perimeter and area for each rectangle

12×8
10×10
Rectangle A
P=40, A=96
Rectangle B
P=40, A=100
Compare
Equal P, B>A
Step 1: Rectangle A: P = 2(12 + 8) = 40 cm, A = 12 × 8 = 96 cm²
Step 2: Rectangle B: P = 2(10 + 10) = 40 cm, A = 10 × 10 = 100 cm²
Step 3: Compare: Same perimeter, but B has larger area
Same perimeter (40 cm), B has larger area (100 vs 96 cm²)
Final answer:

Both rectangles have the same perimeter (40 cm), but Rectangle B has a larger area (100 cm² vs 96 cm²).

Applied rules:

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

• Rectangle area formula: A = lw

• Comparison: Calculate each separately

Rectangle Perimeter and Area Formulas and Properties
P = 2(l + w) and A = l × w
Rectangle Formulas
Perimeter
P = 2(l + w)
Distance around the rectangle
Area
A = l × w
Space inside the rectangle
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
A = l × w
Rectangle Area Formula
Standard Form
A = l × w
Area equals length times width
Square Form
A = s²
Area of square equals side squared
Opposite sides
Equal
Angles
Right (90°)
Diagonals
Equal
Symmetry
2 axes
Key concept: Perimeter measures distance around, area measures space inside.
Unit difference: Perimeter in linear units (cm, m), area in square units (cm², m²).
Square connection: A square is a special rectangle with equal length and width.
Essential Rectangle Rules:

• Perimeter rule: P = 2(l + w)

• Area rule: A = l × w

• Square perimeter: P = 4s

• Square area: A = s²

• Find length: l = (P/2) - w or l = A/w

• Find width: w = (P/2) - l or w = A/l

Detailed Summary: Rectangle Perimeter and Area Geometry
P = 2(l + w) and A = l × w
Core Rectangle Formulas
Key definitions:

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

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

Area of rectangle: The product of length and width (l × w)

Rectangle perimeter and area 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 perimeter, A = l × w for area
  4. Substitute values: Replace l and w with actual measurements
  5. Calculate: Follow order of operations (parentheses first)
  6. Check units: Perimeter in linear units, area in square units
  7. Verify answer: Check if answer makes sense in context
Tip 1: Always identify length and width before applying formulas.
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, length = width, so P = 4s and A = s².
Tip 5: When finding missing dimensions, use algebra to isolate variables.
Tip 6: Always include correct units in your final answer.
Common errors: Confusing perimeter and area, mixing units, calculation mistakes, misidentifying length vs. width.
Problem-solving strategy: Draw the rectangle, label dimensions, apply formulas, check answer.
Rectangle Formulas to Remember:

• Rectangle perimeter: P = 2(l + w)

• Rectangle area: A = l × w

• Square perimeter: P = 4s

• Square area: A = s²

• Find length: l = (P/2) - w or l = A/w

• Find width: w = (P/2) - l or w = A/l

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

Rectangle: l=7, w=2 → P = 2(7+2) = 18, A = 7×2 = 14

Advanced Examples

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

Find length: A=48, w=6 → l = 48/6 = 8

Decimal rectangle: l=5.5, w=3.2 → P = 2(5.5+3.2) = 17.4, A = 5.5×3.2 = 17.6

Key Geometric Concepts:

Perimeter vs Area: Perimeter measures the boundary (distance around), while area measures the interior space (coverage).

Unit Differences: Perimeter uses linear units (cm, m, in), area uses square units (cm², m², in²).

Rectangle Properties: Four right angles, opposite sides equal, diagonals equal, two lines of symmetry.

Problem-Solving Strategies:

• Read carefully: Identify whether the problem asks for perimeter, area, or both

• Draw a picture: Sketch the rectangle and label dimensions

• Choose the right formula: P = 2(l + w) for perimeter, A = lw for area

• Check units: Ensure all measurements are in the same unit

• Verify answer: Does the answer make sense in the context?

Questions & Answers

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

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

  • Perimeter words: "around," "fence," "border," "edge," "distance around," "outline," "boundary"
  • Area words: "inside," "cover," "floor," "surface," "space," "fill," "carpet," "paint"

Think of it this way: If you're walking around the edge of something, you're measuring perimeter. If you're covering the inside with paint or carpet, you're measuring area. For example, fencing needs perimeter while flooring needs area!

Question: My child is struggling with the difference between perimeter and area. How can I help them understand?

Answer: Here are effective strategies to help your child understand perimeter vs. area:

  • Hands-on activities: Use string to measure perimeter of books, tiles to cover area
  • Visual aids: Draw rectangles and shade the area inside, trace the perimeter outside
  • Real examples: Fencing around a yard (perimeter), grass growing inside (area)
  • Physical demonstration: Walk around the edge (perimeter), sweep the inside (area)

The key concept is that perimeter measures the distance around the outside edge, while area measures the space inside the shape. Practice with real objects to make it concrete!

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

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 formulas P = 2(l + w) and A = lw because addition and multiplication are commutative! As long as you have both measurements, you'll get the right answer.