5. Area of Trapezoids

Session 5 Learning Plan

Learning objective: Calculate trapezoid area from two parallel bases and height.

Key vocabulary: trapezoid, parallel bases, height, average

Success criterion: Apply the parentheses and factor \(\tfrac12\) in the correct order.

Learn and Practice

A trapezoid has two parallel sides of different lengths, called \(b_1\) and \(b_2\). Because the two bases differ, you use their average times the height:
\[ A = \tfrac{1}{2}(b_1 + b_2) \times h. \]

Worked example: a trapezoid with parallel sides 6 cm and 10 cm and height 4 cm has area
\[ A = \tfrac{1}{2}(6 + 10) \times 4 = \tfrac{1}{2}(16)(4) = 32 \text{ cm}^2. \]

Reading the formula in words: add the two parallel sides, cut the result in half, then multiply by the height.

Case study - a garden plot shaped like a trapezoid with parallel edges of 8 m and 12 m and a depth of 5 m covers
\[ \tfrac{1}{2}(8 + 12)(5) = \tfrac{1}{2}(20)(5) = 50 \text{ m}^2. \]

Common mistake: forgetting to add the bases before halving. Do the parentheses first: add \(b_1 + b_2\), then take half.

Understanding the Trapezoid Formula

A trapezoid has one pair of parallel sides called bases, \(b_1\) and \(b_2\). Its height is the perpendicular distance between them. The area formula is
\[ A=\tfrac12(b_1+b_2)h. \]
Another useful form is
\[ A=\left(\frac{b_1+b_2}{2}\right)h. \]
The expression in parentheses is the average length of the two bases. A trapezoid therefore has the same area as a rectangle whose width is the average base and whose height is the trapezoid's height.

To see why, duplicate the trapezoid and rotate the copy. Together they form a parallelogram with base \(b_1+b_2\) and height \(h\). One trapezoid is half of that area, giving \(\tfrac12(b_1+b_2)h\).

Identify the Parallel Sides First

Do not assume the top and bottom are the bases merely because of page orientation. The bases are the parallel sides. In a rotated trapezoid they may appear left and right. Parallel-arrow markings or the diagram description identify them. The height must cross from one parallel side to the other at \(90^\circ\).

For bases \(9\text{ cm}\) and \(15\text{ cm}\), with height \(7\text{ cm}\),
\[ A=\tfrac12(9+15)(7)=\tfrac12(24)(7)=84\text{ cm}^2. \]
An efficient route is to average the bases first: \((9+15)/2=12\), then \(12(7)=84\).

Common Calculation Traps

The formula's grouping matters. Add the bases before multiplying by one-half. The incorrect expression \(\tfrac12b_1+b_2h\) does not treat the two bases together. Another trap is using a slanted leg as height. A third is forgetting one base and calculating triangle area.

Use a verbal script: “Add the parallel bases, divide by two, multiply by perpendicular height.” This sentence mirrors the formula and prevents order errors.

Decomposition Verification

A trapezoid can sometimes be split into a rectangle and one or two triangles. Suppose the bases are \(6\) and \(14\), and the height is \(5\). The formula gives
\[ A=\tfrac12(6+14)(5)=50. \]
If the central rectangle is \(6\)-by-\(5\), its area is \(30\). The remaining triangular regions together have a total base of \(14-6=8\), so their combined area is \(\tfrac12(8)(5)=20\). Total: \(30+20=50\). Agreement between methods verifies the result.

Multi-Step Example: Landscaping

A trapezoidal flower bed has parallel sides \(12\) feet and \(20\) feet with perpendicular depth \(9\) feet. Mulch covers \(36\text{ ft}^2\) per bag. How many bags are required?

Find area:
\[ A=\tfrac12(12+20)(9)=\tfrac12(32)(9)=144\text{ ft}^2. \]
Convert area to bags:
\[ 144\div36=4. \]
The bed requires \(4\) bags. If each bag cost \(\$8.50\), total cost would be \(4(8.50)=\$34.00\). Keep each stage labeled so area, bag count, and cost do not become confused.

Work Backward From Area

Suppose \(A=72\text{ m}^2\), \(b_1=10\text{ m}\), \(b_2=14\text{ m}\), and height is unknown:
\[ 72=\tfrac12(10+14)h=12h. \]
Thus \(h=6\text{ m}\). Verify: the average base is \(12\), and \(12(6)=72\). The missing height is a linear measurement, so its unit is meters.

Reasonableness Bound

A trapezoid's average base lies between its shorter and longer base. Therefore its area must lie between \(b_{\text{short}}h\) and \(b_{\text{long}}h\). With bases \(8\) and \(14\), height \(5\), the area must be between \(40\) and \(70\). The exact area \(55\) fits. This quick bound can eliminate answer choices before exact computation.

Session Checkpoint

You are ready to continue when you can identify parallel bases in any orientation, explain the average-base interpretation, use parentheses correctly, verify by decomposition, solve a coverage application, and bound the answer between two rectangle areas.

GED® Coaching: Build a Repeatable Solution

The GED® rewards a dependable process more than memorizing isolated tricks. On every area problem, pause before touching the calculator. Read the final question first and underline the requested quantity. Then label the figure with only the measurements that matter. If the drawing contains extra numbers, ask what role each number plays in the formula. Write the formula in symbols, substitute the measurements with their units, and calculate. This order makes your work easy to check and prevents a calculator entry from becoming an unexplained guess.

Use the GED® formula sheet as a tool, not as a substitute for understanding. The sheet can remind you that a triangle uses \(\tfrac12 bh\), but you must decide which segment is the base and which is the perpendicular height. When a problem is unfamiliar, sketch a simpler version. A rough rectangle, circle, or set of component shapes often reveals the correct operation faster than rereading a dense paragraph.

Estimation is your built-in error detector. Round the dimensions to friendly values and predict the approximate size of the answer. If a rectangle is about \(10\) by \(5\), its area should be near \(50\), not \(5\) or \(500\). Check the unit too: area requires square units. A choice with feet instead of square feet may be testing whether you understand the target rather than whether you can multiply.

Build a visual record as you solve. Shade the region whose area is requested, circle measurements that belong in the formula, and cross out measurements that are irrelevant. For composite figures, draw the cut lines yourself and give each component a short name such as \(A_1\) or “window.” For coordinate figures, write each derived length beside its segment. These small annotations reduce working-memory load and leave evidence you can inspect if the answer looks wrong. They also make it easier to resume a marked problem during a second test pass.

When checking an answer choice, work backward conceptually: ask what calculation could have produced it. If a distractor equals the sum of two dimensions, it probably represents a perimeter-related mistake. If a circle answer is four times your result, diameter may have been used as radius. Recognizing the source of a distractor should confirm sound work, never replace it.

After practice, do not merely record a score. For every miss, write one sentence naming the error and one sentence describing the prevention habit. Examples include: “I used diameter as radius; next time I will mark \(r=d/2\),” or “I added the rectangles twice; next time I will shade non-overlapping parts.” This correction routine turns mistakes into durable GED® skills.

Session Practice Routine

1. State what measurement the problem requests.
2. Mark only the dimensions needed for the selected formula.
3. Write the formula before substituting values.
4. Show the calculation and attach a squared unit.
5. Check the answer by estimation or substitution.

Exit Reflection

Without looking back, write the session's key formula or strategy, name one common error, and explain how you will detect that error on a GED® question.