Area: Derivation and Spatial Application Answer Key

Worksheet 4 of 5 in the Grade 7 circles series. See where $A = \pi r^2$ comes from, then use it for sprinklers, rugs, and pizzas. Next: Advanced Algebra and Partial Figures.

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Lesson Notes

The area of a circle is the amount of flat space inside it, measured in square units ($\text{cm}^2$, $\text{ft}^2$, $\text{m}^2$).

$A = \pi r^2$$r^2$ means $r \times r$

Where the formula comes from

Cut a circle into thin wedges, like pizza slices, and lay them side by side, alternating point-up and point-down. The wedges fit together into a shape that is almost a parallelogram. The more wedges you use, the closer it gets to a true rectangle.

Half the wedges point up, half point down. The curved edges become the top and bottom.

The height of the shape is the radius, $r$.
The base is half of the circumference (the other half is on top): $\tfrac{1}{2} \times 2\pi r = \pi r$.
Area of a parallelogram $= \text{base} \times \text{height} = \pi r \times r = \pi r^2$.

Using the formula

Worked example 1: given the radius

$r = 4$ m. $A = \pi r^2 = 3.14 \times 4 \times 4 = 3.14 \times 16 = 50.24\ \text{m}^2$

Worked example 2: given the diameter

$d = 10$ in. First $r = 10 \div 2 = 5$ in. Then $A = 3.14 \times 25 = 78.5\ \text{in}^2$

Two common mistakes. Square only the radius, not $\pi r$. And if the problem gives a diameter, halve it before you square it: $(d \div 2)^2$, not $d^2 \div 2$.

25 problems. Work each one on the canvas under it (pen, eraser and clear are in the canvas toolbar; drag the handle below the canvas for more room). Enter your answer in the box (the keypad under it has fractions, exponents, square roots, pi, the degree sign and the symbols < > ≤ ≥ ≠). When a Unit row is shown, tap the unit for your answer. Then press Check. The eye at the right of the row shows or hides the worked solution.