Lesson 75: Area of a Complex Figure, Area of a Trapezoid

New Concept

To find the area of a trapezoid, we use the average of its parallel bases and its height. The textbook provides two equivalent formulas for this.

Area of a Trapezoid

What’s next

Next, you'll apply these formulas in worked examples involving both trapezoids and more complex figures. We will also practice estimation strategies for quick calculations.

Property

To find the area of a complex figure, divide it into simpler shapes (like rectangles and triangles) and add their individual areas. Alternatively, enclose the figure in a larger rectangle and subtract the areas of the parts that are not included.

Examples

• Method 1 (Addition): Divide the figure into a $5 \times 8$ rectangle and a triangle with base 4 and height 8. Area = $(5 \times 8) + \frac{1}{2}(4 \times 8) = 40 + 16 = 56 \text{ units}^2$.
• Method 2 (Subtraction): Enclose the figure in a $12 \times 12$ square and subtract a $5 \times 8$ triangle. Area = $(12 \times 12) - \frac{1}{2}(5 \times 8) = 144 - 20 = 124 \text{ units}^2$.

Explanation

To find the area of a funky-shaped figure, slice it up into familiar shapes like rectangles and triangles. Calculate the area of each piece, then add them all up for the grand total! Alternatively, you can frame the figure inside a big rectangle and subtract the areas of the leftover bits you don't need.

Property

To find the area of a trapezoid, multiply the average of the bases by the height. The formula is:

Examples

• A trapezoid has bases $b_1 = 8$ cm and $b_2 = 12$ cm, and height $h = 5$ cm. Area = $\frac{1}{2}(8 + 12) \times 5 = \frac{1}{2}(20) \times 5 = 50 \text{ cm}^2$.
• Find the area of a trapezoid with bases of 11 m and 15 m and a height of 8 m. Area = $\frac{1}{2}(11 + 15) \times 8 = \frac{1}{2}(26) \times 8 = 104 \text{ m}^2$.

Explanation

A trapezoid has two different parallel sides, so which one do we use? Neither! The trick is to find the average of the two bases, then multiply that average by the height. This clever shortcut transforms the trapezoid into a simple rectangle with the same area, making the calculation super straightforward and avoiding any guesswork.

Property

To estimate the area of a figure, first round its dimensions (like base and height) to the nearest convenient numbers. Then, use these rounded numbers to perform the area calculation for a quick approximation.

Examples

• Estimate the area of a trapezoid with bases $4.9$ ft and $8.2$ ft and height $5.8$ ft. Round to bases 5 and 8, height 6. Area $\approx \frac{1}{2}(5 + 8) \times 6 = 39 \text{ ft}^2$.
• Estimate the area of a book cover with parallel sides $3 \frac{7}{8}$ in. and $6 \frac{1}{8}$ in. and a height of 4 in. Round bases to 4 and 6. Area $\approx \frac{1}{2}(4 + 6) \times 4 = 20 \text{ in}^2$.

Explanation

Why wrestle with tricky fractions or decimals when a good guess will do? For a quick estimate, round each dimension—like the bases and height of a trapezoid—to the nearest whole number or an easy-to-use value. Then, plug these friendly numbers into the area formula. You'll get a speedy, 'close-enough' answer without the headache.