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. Show your work in the space under each problem, then put a box around your answer. On screen, the eye at the right of each problem shows or hides the worked solution.