Why you average the two bases
Here’s how it works: flip a copy of your trapezoid upside down and set it beside the first one.
The two copies fit together into a parallelogram. Its base is b₁ + b₂, and its height is still h.
A parallelogram’s area is base times height, so the pair covers h × (b₁ + b₂). Your trapezoid is one of the two copies, so it covers half of that: ½h(b₁ + b₂).
Half of b₁ + b₂ is also the average of the two bases. Bases of 6 and 10 feet average to 8 feet. So a trapezoid 5 feet tall covers the same space as an 8-by-5 rectangle: 40 square feet.
Bottom line: a trapezoid covers the same space as a rectangle as wide as its average base.
Example 1: a wooden deck
A wooden deck is shaped like a trapezoid. The side along the house is 24 feet, and the outer edge is 12 feet. The deck sticks out 8 feet from the house, and each slanted side is 10 feet. What is the area of the deck?
The working
- Write the formula: A = ½h(b₁ + b₂)
- Find the bases. They’re the two parallel sides: 12 feet and 24 feet.
- Find the height. It’s the 8 feet straight out from the house. The 10 feet is a slanted side, so leave it out.
- Add the bases first: 12 + 24 = 36
- Multiply by the height and take half: ½ × 8 × 36 = 144
- Add square units. Area is always in square units, so it’s 144 square feet.
Answer: 144 square feet
Check it: the deck covers more than a 12-by-8 rectangle (96 square feet) and less than a 24-by-8 one (192). The 144 sits right in the middle.
Example 2: work backward to the height
Some questions give you the area and ask for the height. Use the same formula and solve for h.
A garden bed shaped like a trapezoid covers 54 square feet. Its two parallel sides are 7 feet and 11 feet. How far apart are they?
The working
- Write the formula: A = ½h(b₁ + b₂)
- Put in what you know: 54 = ½ × h × (7 + 11)
- Add the bases and take half: 7 + 11 = 18, and half of 18 is 9. So 54 = 9 × h
- Divide both sides by 9: h = 54 ÷ 9 = 6
- Check it: ½ × 6 × 18 = 54. The sides are 6 feet apart.
Answer: 6 feet
Example 3: area of a trapezoid without the height
The catch? Some questions leave out the height and give you the slanted sides instead. You can still find the height with the Pythagorean theorem, a² + b² = c², which is also on the formula sheet.
One side of a roof is a trapezoid. Its top edge is 20 feet, and its bottom edge is 30 feet. Each slanted edge is 13 feet. How many square feet of shingles does it need?
The working
- Find how far each end sticks out. The bottom is 30 − 20 = 10 feet longer, so each end sticks out 10 ÷ 2 = 5 feet.
- The slanted edge, the height and the 5 feet make a right triangle. The slanted edge is the longest side: h² + 5² = 13²
- Solve for h: h² = 169 − 25 = 144, so h = √144 = 12
- Use the area formula: A = ½ × 12 × (20 + 30) = 6 × 50 = 300
- Add square units: 300 square feet.
Answer: 300 square feet
Splitting the 10 feet in half works because both slanted edges are the same length. Then each end sticks out the same amount.
4 mistakes that cost points
Mistake 1: using the slanted side as the height
It’s the same trap as in the parallelogram lesson. Use the deck’s 10-foot slanted side and you get 180 square feet. The straight-out 8 feet gives the right 144.
Mistake 2: forgetting the ½
Add the bases and multiply by the height, and you’ve found the area of two trapezoids. For the deck, that’s 288 square feet, twice the real answer.
Mistake 3: multiplying only one base
Type 0.5 × 8 × 12 + 24 into a calculator, and only the 12 gets multiplied. You get 72 square feet, half the real answer. Add the bases first, or type the parentheses.
Mistake 4: mixing feet and inches
Change every length to the same unit first. Say a window’s parallel sides are 2 feet and 30 inches, and it’s 18 inches tall. Use 24 inches for the 2 feet: ½ × 18 × (24 + 30) = 486 square inches.
Practice: 5 new questions
Every set is new. Check each answer as you go, and read the working on any you miss before the next one.
Area of a trapezoid: 5 questions
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New questions every time you start. Do this set as many times as you like.
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Want more? Practice all of geometry and measurement (25 questions, every skill in the topic), or take the full GED math practice test.
Frequently asked questions
What are b1 and b2 in a trapezoid?
They're the two bases: the parallel sides, usually drawn as the top and the bottom. The small 1 and 2 only tell them apart. It doesn't matter which one you call b₁, because you add them anyway: 9 + 15 and 15 + 9 are both 24. The slanted sides are never bases.
What is the area of a trapezoid with bases of 8 cm and 12 cm and a height of 5 cm?
50 square centimeters. Add the bases: 8 + 12 = 20. Multiply by the height: 20 × 5 = 100. Then take half: 100 ÷ 2 = 50. You can also average the bases first. Halfway between 8 and 12 is 10, and 10 × 5 = 50 square centimeters too.
Keep going
Area of a parallelogram
Base times height, and how to spot the slanted side that isn't the height.
Volume of a cylinder
The next lesson: the circle on the bottom times the height, plus gallons.
GED math formula sheet
Every formula on the official sheet with a real-life example, plus the 7 it leaves out.
Free printable practice test
Two full tests on paper, with a worked solution for all 92 questions.
Written by
Coach Crissy
Math instructor
Coach Crissy teaches the video lessons behind the BooknBoost GED math course, working through GED-style problems on camera: ratios and proportions, percent and interest, mean, median and mode, slope, and the geometry students meet on test day.