A trapezoid has parallel sides of 5 cm and 9 cm, slanted legs of 6 cm each, and a perpendicular height of 4 cm. What is its area?
A56 cm² — (5 + 9) × 4
B28 cm² — (1/2)(5 + 9) × 4
C24 cm² — (1/2)(6 + 6) × 4, using the slanted legs instead of the parallel sides
D18 cm² — (1/2)(9) × 4, using only the longer base
The correct answer is 28 cm². The formula is A = (1/2)(b₁ + b₂)h, where b₁ and b₂ are the two parallel sides: (1/2)(5 + 9)(4) = (1/2)(14)(4) = 28. Option A forgets the essential 1/2 factor. Option C uses the slanted legs (6 cm each) instead of the parallel sides — the most common conceptual error, since the legs are irrelevant to area. Option D uses only one base instead of adding both.
Question 2 Multiple Choice
Why does the trapezoid area formula include a factor of 1/2?
ABecause the two bases must be averaged by dividing their sum by 2 before multiplying by height
BBecause two identical trapezoids joined together form a parallelogram with area (b₁ + b₂)h, so each trapezoid is exactly half of that
CBecause the perpendicular height is always half the length of the slant height
DBecause only half of each base contributes to the enclosed interior area
The derivation: flip a copy of the trapezoid upside down and attach it to the original along one base. The result is a parallelogram with base (b₁ + b₂) and height h, whose area is (b₁ + b₂)h. Since two trapezoids make one parallelogram, one trapezoid is half: A = (1/2)(b₁ + b₂)h. Option A restates the formula's arithmetic equivalence (average of bases times height) but doesn't explain where the 1/2 comes from. Options C and D describe fictional geometric properties.
Question 3 True / False
In the formula A = (1/2)(b₁ + b₂)h, b₁ and b₂ can represent any two sides of the trapezoid as long as they are the longest two sides.
TTrue
FFalse
Answer: False
b₁ and b₂ must specifically be the two parallel sides — not the longest sides, not the slanted legs, and not any other pair. The formula only works because the parallel sides define the trapezoid's width at the top and bottom, and the perpendicular height measures the distance between them. Using the slanted legs instead is the most common area calculation error on trapezoid problems.
Question 4 True / False
A parallelogram can be seen as a special case of a trapezoid where both parallel sides are equal, and applying the trapezoid formula to it produces the standard parallelogram area formula A = bh.
TTrue
FFalse
Answer: True
When b₁ = b₂ = b, the formula becomes A = (1/2)(b + b)h = (1/2)(2b)h = bh — exactly the parallelogram formula. Similarly, when one base shrinks to zero, (1/2)(b + 0)h = (1/2)bh — the triangle formula. This shows all three formulas are the same formula in different conditions, and every polygon area can be understood through decomposition and rearrangement.
Question 5 Short Answer
Describe the 'doubling' derivation of the trapezoid area formula. What shape results when two identical trapezoids are combined, and how does this explain both the (b₁ + b₂) term and the 1/2 factor?
Think about your answer, then reveal below.
Model answer: Flip a copy of the trapezoid upside down and attach it to the original along one of its parallel sides. The result is a parallelogram whose base equals the sum of the two trapezoid bases (b₁ + b₂) and whose height equals the trapezoid's height h. This parallelogram has area (b₁ + b₂)h. Since two trapezoids make up this parallelogram, one trapezoid is half: A = (1/2)(b₁ + b₂)h.
The derivation works because every trapezoid can be paired with its mirror image to form a parallelogram with a clean area formula. The (b₁ + b₂) term arises because the top of one trapezoid and the bottom of its flipped copy together form the full base of the parallelogram. The 1/2 reflects that we want only one of the two trapezoids. This connection to the parallelogram formula shows the area formulas for parallelograms, trapezoids, and triangles as a unified family.