Solved Exercises on Area of Parallelograms in Grade 7

Master area of parallelograms: base-height method, special cases, and composite figures through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Parallelogram Area
Exercise 1
Find the area of a parallelogram with base 15 cm and height 8 cm.
Definition:

Height of a Parallelogram: The perpendicular distance between opposite sides (the base and the side parallel to it).

Method for parallelogram area:
  1. Identify the base and corresponding height
  2. Apply the area formula: A = base × height
  3. Substitute the values
  4. Calculate the result
  5. Include the units (cm²)
Given
Base = 15 cm, Height = 8 cm
Formula
A = bh
Result
120 cm²
Step 1: Identify the base and height

Base = 15 cm, Height = 8 cm

Step 2: Apply the area formula

Area = base × height

Step 3: Substitute values

Area = 15 × 8

Step 4: Calculate

Area = 120 cm²

Step 5: Verify the calculation

Check: 15 × 8 = 120 ✓

Area = 120 cm²
Final answer:

The area of the parallelogram is 120 cm².

Applied rules:

• Parallelogram Area Formula: A = base × height

• Perpendicular Height: Height must be perpendicular to base

• Units: Area units are squared (cm²)

2 Finding Missing Base or Height
Exercise 2
The area of a parallelogram is 96 cm² and its height is 12 cm. Find the length of the base.
Definition:

Rearranging the Area Formula: When area and one dimension are known, rearrange A = bh to find the missing dimension.

Given
Area = 96 cm², Height = 12 cm
Formula
A = bh → b = A/h
Result
8 cm
Step 1: Start with the area formula

A = base × height

Step 2: Rearrange to solve for base

base = Area ÷ height

Step 3: Substitute known values

base = 96 ÷ 12

Step 4: Calculate

base = 8 cm

Step 5: Verify the solution

Area = 8 × 12 = 96 cm² ✓

Base = 8 cm
Final answer:

The length of the base is 8 cm.

Applied rules:

• Formula Rearrangement: Isolate the unknown variable

• Algebraic Manipulation: Divide both sides by height

• Verification: Substitute back to check the answer

3 Special Case - Rectangle
Exercise 3
Find the area of a rectangle with length 10 cm and width 7 cm. Explain how this relates to parallelogram area.
Definition:

Rectangle: A special type of parallelogram where all angles are 90°. The area formula remains A = base × height.

Given
Length = 10 cm, Width = 7 cm
Formula
A = lw (or bh)
Result
70 cm²
Step 1: Recognize that a rectangle is a parallelogram

Rectangle is a special case of parallelogram with all angles equal to 90°

Step 2: Apply the area formula

Area = length × width (which is base × height)

Step 3: Substitute values

Area = 10 × 7

Step 4: Calculate

Area = 70 cm²

Step 5: Explain the relationship

In a rectangle, the length and width are perpendicular to each other, so width = height

Thus: A = length × width = base × height

Area = 70 cm²
Final answer:

The area of the rectangle is 70 cm². A rectangle is a special parallelogram where the height equals the width.

Applied rules:

• Rectangle Property: Special case of parallelogram

• Area Formula: A = base × height (same for both)

• Perpendicular Sides: In rectangle, width serves as height

Parallelogram Area Properties and Methods
\(A = \text{base} \times \text{height}\)
Parallelogram Area Formula
Rectangle
A = lw
Special parallelogram
Rhombus
A = bh or A = (d₁d₂)/2
Equal sides parallelogram
Height
h = A/b
From area formula
Key definitions:

Parallelogram: A quadrilateral with opposite sides parallel and equal in length.

Base: Any side of the parallelogram that serves as the reference for measuring height.

Height: The perpendicular distance between the base and the opposite parallel side.

Rectangle: A parallelogram with four right angles.

Rhombus: A parallelogram with four equal sides.

Square: A parallelogram that is both a rectangle and a rhombus.

Complete methodology:
  1. Identify the Parallelogram: Ensure opposite sides are parallel
  2. Determine Base and Height: Ensure height is perpendicular to base
  3. Apply the Formula: A = base × height
  4. Substitute Values: Plug in the known measurements
  5. Calculate: Perform the multiplication
  6. Check Units: Ensure answer is in square units
Tip 1: The height must be perpendicular to the base (at 90°).
Tip 2: Any side can be the base if you use the corresponding perpendicular height.
Tip 3: In rectangles, the adjacent sides serve as base and height.
Tip 4: Always include units in your final answer (cm², m², etc.).
Common errors: Using slanted sides instead of perpendicular height, confusing perimeter with area, misidentifying base and height.
Exam preparation: Practice with different parallelogram orientations, work with fractional measurements, master formula rearrangement.
Formulas to know by heart:

• Parallelogram Area: A = bh

• Base: b = A/h

• Height: h = A/b

• Rectangle Area: A = length × width

• Rhombus Area: A = bh or A = (d₁d₂)/2

• Square Area: A = s²

Solution: Exercises 4 to 5
4 Special Case - Rhombus
Exercise 4
A rhombus has diagonals measuring 12 cm and 16 cm. Find its area using both methods: base-height and diagonal formula.
Definition:

Rhombus: A parallelogram with four equal sides. It has two area formulas: A = bh and A = (d₁d₂)/2.

Given
d₁ = 12 cm, d₂ = 16 cm
Diagonal Formula
A = (d₁d₂)/2
Result
96 cm²
Step 1: Calculate area using diagonal formula

Area = (d₁ × d₂) ÷ 2

Area = (12 × 16) ÷ 2 = 192 ÷ 2 = 96 cm²

Step 2: Find the side length using diagonals

Diagonals of a rhombus bisect each other at right angles

Half-diagonals: 6 cm and 8 cm

Using Pythagorean theorem: s² = 6² + 8² = 36 + 64 = 100

Side length: s = 10 cm

Step 3: Find the height using area

Area = base × height

96 = 10 × height

Height = 96 ÷ 10 = 9.6 cm

Step 4: Verify using base-height formula

Area = base × height = 10 × 9.6 = 96 cm² ✓

Area = 96 cm² (both methods)
Final answer:

The area of the rhombus is 96 cm², verified by both formulas.

Applied rules:

• Rhombus Diagonal Formula: A = (d₁d₂)/2

• Diagonal Property: Diagonals bisect at right angles

• Pythagorean Theorem: Used to find side length

5 Composite Figure
Exercise 5
A rectangular garden measuring 20 m by 15 m has a parallelogram-shaped flower bed cut out from the center. The flower bed has a base of 8 m and a height of 5 m. Find the remaining area of the garden.
Definition:

Composite Figure: A shape made up of two or more simple geometric figures. To find the area, calculate the area of each part and combine as needed.

Rectangle
20m × 15m = 300m²
Parallelogram
8m × 5m = 40m²
Remaining
300 - 40 = 260m²
Step 1: Calculate the area of the rectangle

Rectangle Area = length × width

Rectangle Area = 20 × 15 = 300 m²

Step 2: Calculate the area of the parallelogram flower bed

Parallelogram Area = base × height

Parallelogram Area = 8 × 5 = 40 m²

Step 3: Subtract the flower bed area from the rectangle area

Remaining Area = Rectangle Area - Flower Bed Area

Remaining Area = 300 - 40 = 260 m²

Step 4: Verify the calculation

Check: 300 - 40 = 260 ✓

The remaining area should be less than the original area ✓

Remaining area = 260 m²
Final answer:

The remaining area of the garden is 260 m².

Applied rules:

• Composite Area: Total Area = Sum of parts or Difference of parts

• Rectangle Area: A = length × width

• Parallelogram Area: A = base × height

Complete Guide: Area of Parallelograms
\(\text{Height} = \frac{\text{Area}}{\text{Base}}\)
Height from Area Formula
Key definitions:

Parallelogram: A quadrilateral with opposite sides parallel and equal in length. Opposite angles are also equal.

Area: The measure of the surface enclosed by a shape, expressed in square units (cm², m², etc.).

Base: The side of a parallelogram that is used as a reference for measuring the height.

Height: The perpendicular distance from the base to the opposite parallel side.

Rhombus: A parallelogram with four equal sides.

Rectangle: A parallelogram with four right angles.

Square: A parallelogram that is both a rectangle and a rhombus.

Complete methodology:
  1. Identify the Parallelogram: Confirm it has opposite sides parallel
  2. Find Base and Height: Ensure the height is perpendicular to the base
  3. Select Appropriate Formula: Use A = base × height
  4. Substitute Values: Replace variables with given measurements
  5. Calculate Carefully: Perform multiplication accurately
  6. Verify Solution: Check calculations and ensure answer is reasonable
Tip 1: The height of a parallelogram is always perpendicular to the base, not necessarily a side of the parallelogram.
Tip 2: You can use any side as the base as long as you use the corresponding perpendicular height.
Tip 3: In rectangles, the length and width serve as base and height.
Tip 4: Always check that your answer makes sense by comparing it to the given dimensions.

Common errors: Using slanted sides instead of perpendicular height, confusing perimeter with area, misidentifying base-height pairs, incorrect unit conversion.
Exam preparation: Practice with parallelograms in different orientations, master formula rearrangement, work with decimal and fractional measurements.
Formulas to know by heart:

• Standard Parallelogram: A = bh

• Base: b = A/h

• Height: h = A/b

• Rectangle: A = length × width

• Rhombus: A = bh or A = (d₁d₂)/2

• Square: A = s²

• Area with Trigonometry: A = ab sin(θ)

Visualizing Parallelogram Areas: Different Base-Height Combinations
Exercise 6: Area Relationships
Consider how area changes with different base-height combinations:
Fixed area of 60 m² with varying base and height values
Showing the inverse relationship

Analysis: The chart shows how base and height are inversely related for a fixed area.

  • When base doubles, height halves to maintain the same area
  • Product of base and height is constant for fixed area
  • Area remains the same regardless of base-height combination
  • The relationship follows bh = A (constant)

Questions & Answers

Question: How do I find the height of a parallelogram if it's not given and doesn't seem to be one of the sides?

Answer: This is a common challenge! Here are several approaches:

When height is not obvious:

  • Draw the altitude: Sketch a perpendicular line from any point on one base to the opposite parallel side
  • Rearrange the area formula: If area is known, use h = A/b
  • Use trigonometry: If an angle and side are known, use sine function
  • Pythagorean theorem: If you can form a right triangle with the height

For tilted parallelograms: The height may lie outside the shape, so extend the base if necessary to measure the perpendicular distance.

Remember: The height is always perpendicular to the base, regardless of the parallelogram's orientation.

Question: Can I use any side of a parallelogram as the base? Will the area be the same?

Answer: Yes, you can use any side as the base, and the area will always be the same! Here's why:

Key principle:

  • The area of a parallelogram is constant regardless of which side you consider the base
  • Whichever side you choose as the base, you must use the corresponding perpendicular height
  • The product of base and height will always equal the area

Example: In a parallelogram with sides 6 and 10, and heights 8 and 4.8 respectively:

  • Using base = 6: Area = 6 × 8 = 48
  • Using base = 10: Area = 10 × 4.8 = 48

This demonstrates that area is invariant under different base choices.

Question: Why does the parallelogram area formula work? Why is it the same as rectangle area?

Answer: The formula works due to the relationship between parallelograms and rectangles:

Visual explanation:

  • If you cut off a triangle from one end of a parallelogram and move it to the other end, you create a rectangle
  • The area remains unchanged during this transformation
  • The rectangle has the same base and height as the original parallelogram
  • Rectangle area = base × height
  • Therefore, Parallelogram area = base × height

Alternative perspective:

You can think of a parallelogram as a "slanted" rectangle. The perpendicular height determines how much vertical space the shape occupies, regardless of the slant.