Solved Exercises on Area of Trapezoids in Grade 7

Master area of trapezoids: parallel sides method, special cases, and composite figures through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Trapezoid Area
Exercise 1
Find the area of a trapezoid with parallel sides of 10 cm and 14 cm, and a height of 6 cm.
Definition:

Parallel Sides (Bases): The two sides of a trapezoid that are parallel to each other. These are called the top base and bottom base.

Method for trapezoid area:
  1. Identify the two parallel sides (bases)
  2. Identify the height (perpendicular distance between bases)
  3. Apply the area formula: A = ½(b₁ + b₂)h
  4. Substitute the values
  5. Calculate the result
  6. Include the units (cm²)
Given
b₁ = 10 cm, b₂ = 14 cm, h = 6 cm
Formula
A = ½(b₁ + b₂)h
Result
72 cm²
Step 1: Identify the parallel sides and height

b₁ = 10 cm, b₂ = 14 cm, h = 6 cm

Step 2: Apply the area formula

Area = ½ × (sum of parallel sides) × height

Area = ½ × (b₁ + b₂) × h

Step 3: Substitute values

Area = ½ × (10 + 14) × 6

Step 4: Calculate

Area = ½ × 24 × 6

Area = 12 × 6 = 72 cm²

Step 5: Verify the calculation

Check: (10 + 14) ÷ 2 = 12, then 12 × 6 = 72 ✓

Area = 72 cm²
Final answer:

The area of the trapezoid is 72 cm².

Applied rules:

• Trapezoid Area Formula: A = ½(b₁ + b₂)h

• Parallel Sides: Must identify the two parallel sides

• Perpendicular Height: Height must be perpendicular to bases

2 Finding Missing Base
Exercise 2
The area of a trapezoid is 120 cm². If the height is 8 cm and one base is 15 cm, find the length of the other base.
Definition:

Rearranging the Area Formula: When area and some dimensions are known, rearrange A = ½(b₁ + b₂)h to find the missing dimension.

Given
Area = 120 cm², h = 8 cm, b₁ = 15 cm
Formula
A = ½(b₁ + b₂)h → b₂ = (2A/h) - b₁
Result
15 cm
Step 1: Start with the area formula

A = ½(b₁ + b₂)h

Step 2: Rearrange to solve for b₂

2A = (b₁ + b₂)h

2A/h = b₁ + b₂

b₂ = (2A/h) - b₁

Step 3: Substitute known values

b₂ = (2 × 120)/8 - 15

b₂ = 240/8 - 15

b₂ = 30 - 15

Step 4: Calculate

b₂ = 15 cm

Step 5: Verify the solution

Area = ½(15 + 15) × 8 = ½ × 30 × 8 = 15 × 8 = 120 cm² ✓

Other base = 15 cm
Final answer:

The length of the other base is 15 cm.

Applied rules:

• Formula Rearrangement: Isolate the unknown variable

• Algebraic Manipulation: Multiply both sides by 2, then divide by height

• Verification: Substitute back to check the answer

3 Special Case - Rectangle
Exercise 3
Find the area of a rectangle with length 12 cm and width 8 cm using the trapezoid area formula. Explain why this works.
Definition:

Rectangle as Trapezoid: A rectangle is a special case of a trapezoid where both pairs of opposite sides are parallel. Both bases are equal in length.

Given
Length = 12 cm, Width = 8 cm
Trapezoid Formula
A = ½(b₁ + b₂)h
Result
96 cm²
Step 1: Recognize rectangle as special trapezoid

Rectangle is a trapezoid where both bases are equal: b₁ = b₂ = 12 cm

Height = width = 8 cm

Step 2: Apply the trapezoid area formula

Area = ½(b₁ + b₂)h

Step 3: Substitute values

Area = ½(12 + 12) × 8

Area = ½(24) × 8

Area = 12 × 8

Step 4: Calculate

Area = 96 cm²

Step 5: Verify with rectangle formula

Rectangle Area = length × width = 12 × 8 = 96 cm² ✓

Area = 96 cm²
Final answer:

The area of the rectangle is 96 cm². A rectangle is a special trapezoid where both bases are equal.

Applied rules:

• Rectangle Property: Special case of trapezoid

• Area Formula: A = ½(b₁ + b₂)h (becomes A = ½(2b)h = bh)

• Equality of Bases: In rectangle, both parallel sides are equal

Trapezoid Area Properties and Methods
\(A = \frac{1}{2}(b_1 + b_2) \times h\)
Trapezoid Area Formula
Standard
A = ½(b₁+b₂)h
Two parallel sides
Rectangle
A = bh
b₁ = b₂ = length
Midsegment
A = m × h
m = (b₁+b₂)/2
Key definitions:

Trapezoid: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases.

Bases: The two parallel sides of a trapezoid. Usually denoted as b₁ and b₂.

Legs: The two non-parallel sides of a trapezoid.

Height: The perpendicular distance between the two parallel bases.

Isosceles Trapezoid: A trapezoid with equal legs and equal base angles.

Midsegment: The line segment connecting the midpoints of the legs. Its length equals the average of the bases.

Complete methodology:
  1. Identify the Trapezoid: Ensure exactly one pair of sides is parallel
  2. Determine Bases and Height: Identify the two parallel sides and perpendicular height
  3. Apply the Formula: A = ½(b₁ + b₂)h
  4. Substitute Values: Plug in the known measurements
  5. Calculate: Perform the addition, multiplication, and division
  6. Check Units: Ensure answer is in square units
Tip 1: The height must be perpendicular to both bases.
Tip 2: Average of the bases multiplied by height gives the area.
Tip 3: Rectangles and parallelograms are special cases of trapezoids.
Tip 4: Always include units in your final answer (cm², m², etc.).
Common errors: Confusing legs with bases, using slanted sides instead of perpendicular height, misidentifying parallel sides.
Exam preparation: Practice with different trapezoid orientations, work with fractional measurements, master formula rearrangement.
Formulas to know by heart:

• Standard Trapezoid: A = ½(b₁ + b₂)h

• Base: b₂ = (2A/h) - b₁

• Height: h = 2A/(b₁ + b₂)

• Midsegment: m = (b₁ + b₂)/2

• Using Midsegment: A = m × h

• Rectangle: A = length × width

Solution: Exercises 4 to 5
4 Isosceles Trapezoid
Exercise 4
An isosceles trapezoid has bases of 16 cm and 10 cm, and legs of 5 cm each. Find the height and then the area of the trapezoid.
Definition:

Isosceles Trapezoid: A trapezoid with equal legs and equal base angles. The height can be found using the Pythagorean theorem.

Given
b₁ = 16 cm, b₂ = 10 cm, legs = 5 cm
Height
4 cm
Area
52 cm²
Step 1: Understand the trapezoid structure

When we draw the height from both top vertices, we create two right triangles and a rectangle

The base of each right triangle = (16 - 10) ÷ 2 = 3 cm

Step 2: Apply the Pythagorean theorem

In the right triangle: leg² = height² + base²

5² = h² + 3²

25 = h² + 9

h² = 16

h = 4 cm

Step 3: Calculate the area

Area = ½(b₁ + b₂)h

Area = ½(16 + 10) × 4

Area = ½(26) × 4 = 13 × 4 = 52 cm²

Step 4: Verify the calculation

Check: 3² + 4² = 9 + 16 = 25 = 5² ✓

Area check: ½(26) × 4 = 52 ✓

Height = 4 cm, Area = 52 cm²
Final answer:

The height of the trapezoid is 4 cm and the area is 52 cm².

Applied rules:

• Isosceles Trapezoid Property: Equal legs create symmetric triangles

• Pythagorean Theorem: Used to find height from leg length

• Trapezoid Area Formula: A = ½(b₁ + b₂)h

5 Composite Figure
Exercise 5
A garden plot is shaped like a rectangle with a trapezoidal flower bed cut out from one end. The rectangle is 20 m by 12 m. The trapezoid has parallel sides of 8 m and 4 m, with a height of 3 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 × 12m = 240m²
Trapezoid
½(8+4)×3 = 18m²
Remaining
240 - 18 = 222m²
Step 1: Calculate the area of the rectangle

Rectangle Area = length × width

Rectangle Area = 20 × 12 = 240 m²

Step 2: Calculate the area of the trapezoidal flower bed

Trapezoid Area = ½(b₁ + b₂)h

Trapezoid Area = ½(8 + 4) × 3 = ½(12) × 3 = 6 × 3 = 18 m²

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

Remaining Area = Rectangle Area - Trapezoid Area

Remaining Area = 240 - 18 = 222 m²

Step 4: Verify the calculation

Check: 240 - 18 = 222 ✓

The remaining area should be less than the original area ✓

Remaining area = 222 m²
Final answer:

The remaining area of the garden is 222 m².

Applied rules:

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

• Rectangle Area: A = length × width

• Trapezoid Area: A = ½(b₁ + b₂)h

Complete Guide: Area of Trapezoids
\(\text{Height} = \frac{2 \times \text{Area}}{b_1 + b_2}\)
Height from Area Formula
Key definitions:

Trapezoid: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases.

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

Bases: The two parallel sides of a trapezoid, usually denoted as b₁ (top base) and b₂ (bottom base).

Height: The perpendicular distance between the two parallel bases.

Legs: The two non-parallel sides of a trapezoid.

Isosceles Trapezoid: A trapezoid with equal legs and equal base angles.

Midsegment: The segment connecting the midpoints of the legs; its length equals the average of the bases.

Complete methodology:
  1. Identify the Trapezoid: Confirm it has exactly one pair of parallel sides
  2. Find Bases and Height: Ensure height is perpendicular to the bases
  3. Select Appropriate Formula: Use A = ½(b₁ + b₂)h
  4. Substitute Values: Replace variables with given measurements
  5. Calculate Carefully: Perform addition, multiplication, and division in correct order
  6. Verify Solution: Check calculations and ensure answer is reasonable
Tip 1: The height of a trapezoid is always perpendicular to both bases, not necessarily a side of the trapezoid.
Tip 2: You can think of the trapezoid area as the average of the bases multiplied by the height.
Tip 3: In isosceles trapezoids, the legs are equal, which can help find the height using the Pythagorean theorem.
Tip 4: Always check that your answer makes sense by comparing it to the given dimensions.

Common errors: Confusing legs with bases, using slanted sides instead of perpendicular height, misidentifying parallel sides, incorrect unit conversion.
Exam preparation: Practice with trapezoids in different orientations, master formula rearrangement, work with decimal and fractional measurements.
Formulas to know by heart:

• Standard Trapezoid: A = ½(b₁ + b₂)h

• Base 2: b₂ = (2A/h) - b₁

• Base 1: b₁ = (2A/h) - b₂

• Height: h = 2A/(b₁ + b₂)

• Midsegment: m = (b₁ + b₂)/2

• Using Midsegment: A = m × h

• Rectangle: A = length × width

Visualizing Trapezoid 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 sums and heights
Showing the inverse relationship

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

  • When base sum doubles, height halves to maintain the same area
  • Product of (base sum) and height is constant for fixed area
  • Area remains the same regardless of base-height combination
  • The relationship follows (b₁+b₂)h = 2A (constant)

Questions & Answers

Question: How do I find the height of a trapezoid 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 one base to the other parallel base
  • Rearrange the area formula: If area is known, use h = 2A/(b₁ + b₂)
  • Use the Pythagorean theorem: If you can form a right triangle with the height
  • Trigonometry: If an angle and side are known, use sine function

For non-isosceles trapezoids: The height may need to be measured at different positions along the bases.

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

Question: Why does the trapezoid area formula work? What's the intuition behind it?

Answer: The formula has an elegant geometric explanation:

Geometric intuition:

  • If you make a copy of the trapezoid and flip it, you can form a parallelogram
  • The base of this parallelogram is (b₁ + b₂) and the height is the same
  • Parallelogram area = (b₁ + b₂) × h
  • Since this is twice the original trapezoid: Trapezoid area = ½(b₁ + b₂)h

Alternative view:

Think of it as the average of the bases multiplied by the height: A = ((b₁ + b₂)/2) × h. This represents the area of a rectangle with the average width of the trapezoid.

Question: What's the difference between a trapezoid and a trapezium? Do they have the same area formula?

Answer: This is a regional terminology difference:

US Terminology:

  • Trapezoid: Quadrilateral with exactly one pair of parallel sides
  • Trapezium: Quadrilateral with no parallel sides

UK Terminology:

  • Trapezium: Quadrilateral with exactly one pair of parallel sides
  • Trapezoid: Quadrilateral with no parallel sides

In Grade 7 mathematics, we use US terminology where a trapezoid has exactly one pair of parallel sides. The area formula A = ½(b₁ + b₂)h applies to this definition.