Basic Trapezoid Area

Find the area of a trapezoid with bases of 8 cm and 12 cm, and a height of 5 cm.

Write the formula: $A = \frac{(b_1 + b_2) \times h}{2}$ = Area formula identified

Identify the values: $b_1 = 8$ cm, $b_2 = 12$ cm, $h = 5$ cm = Values identified

Add the bases: $b_1 + b_2 = 8 + 12 = 20$ cm = Sum of bases = 20 cm

Multiply by height: $20 \times 5 = 100$ = 100

Divide by 2: $\frac{100}{2} = 50$ = 50 cm²

Answer: The area is $50$ cm²

Trapezoid with Decimal Measurements

A garden bed shaped like a trapezoid has parallel sides of 4.5 m and 6.5 m. The perpendicular distance between them is 3 m. What is the area?

Write the formula: $A = \frac{(b_1 + b_2) \times h}{2}$ = Area formula

Substitute the values: $A = \frac{(4.5 + 6.5) \times 3}{2}$ = Values substituted

Add the bases: $4.5 + 6.5 = 11$ m = Sum = 11 m

Multiply by height: $11 \times 3 = 33$ = 33

Divide by 2: $\frac{33}{2} = 16.5$ = 16.5 m²

Answer: The garden bed has an area of $16.5$ m²

Finding a Missing Dimension

A trapezoid has an area of 42 cm², a height of 6 cm, and one base of 5 cm. Find the other base.

Write the formula and substitute known values: $42 = \frac{(5 + b_2) \times 6}{2}$ = Equation set up

Multiply both sides by 2: $84 = (5 + b_2) \times 6$ = 84 = 6(5 + b₂)

Divide both sides by 6: $\frac{84}{6} = 5 + b_2$ = 14 = 5 + b₂

Subtract 5 from both sides: $14 - 5 = b_2$ = b₂ = 9

State the answer: $b_2 = 9$ cm = The second base is 9 cm

Answer: The other base is $9$ cm

Mistake: Forgetting to divide by 2

Why: The formula requires dividing by 2 because a trapezoid is half of a parallelogram formed by the sum of the bases.

Correct: Always include the division: $A = \frac{(b_1 + b_2) \times h}{2}$, not $(b_1 + b_2) \times h$

Mistake: Using a slant side instead of the height

Why: The height must be perpendicular to both bases. The slanted sides are not the height.

Correct: Always use the perpendicular distance between the parallel sides as the height.

Mistake: Multiplying the bases instead of adding them

Why: The formula requires adding $b_1 + b_2$, not multiplying.

Correct: Remember: ADD the bases first, then multiply by height, then divide by 2.

Roof Cross-Sections

Many roof designs have trapezoidal cross-sections where calculating area helps determine material needs.

A roof section has parallel edges of 6 m (top) and 10 m (bottom), with a height of 2.5 m. Area = $\frac{(6 + 10) \times 2.5}{2} = 20$ m²

Swimming Pool Design

Many swimming pools have a trapezoidal shape when viewed from above to fit irregular yard spaces.

A pool is 8 m at one end, 12 m at the other, and 15 m long (perpendicular distance). Area = $\frac{(8 + 12) \times 15}{2} = 150$ m²

Q: Why do we add the bases and divide by 2?

A: Adding the bases and dividing by 2 gives you the average length of the bases. A trapezoid's area equals this average base times the height, which is why the formula works.

Q: What if the trapezoid is upside down?

A: It doesn't matter which base you call $b_1$ or $b_2$. Since we add them together, the order doesn't affect the result.

Q: How is the trapezoid formula related to other area formulas?

A: If both bases are equal ($b_1 = b_2$), the trapezoid becomes a rectangle, and the formula simplifies to $A = b \times h$. The trapezoid formula is a more general version that works for any quadrilateral with parallel sides.