Advanced Algebra and Partial Figures Answer Key

Worksheet 5 of 5 in the Grade 7 circles series. Work backward from $C$ or $A$, handle semicircles, rings, and shaded regions, and write exact answers in terms of $\pi$. Start of series: Circle Anatomy.

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Lesson Notes

This worksheet combines everything from the first four. The formulas are the same. What changes is that you must decide which formula fits, sometimes solve for a missing piece, and often add or subtract shapes.

$d = 2r$$C = 2\pi r = \pi d$$A = \pi r^2$

Answers in terms of $\pi$

An exact answer keeps the symbol $\pi$ instead of rounding it to 3.14. Treat $\pi$ like a variable: $2 \times \pi \times 4 = 8\pi$, and $9\pi + 16\pi = 25\pi$. Numbers without $\pi$ stay separate: $12 + 6\pi$ cannot be combined any further.

Working backward

From $C$: $C = 2\pi r$, so $r = C \div 2\pi$. If $C = 18\pi$, then $r = 18\pi \div 2\pi = 9$.
From $A$: $A = \pi r^2$, so $r^2 = A \div \pi$, then take the square root. If $A = 25\pi$, then $r^2 = 25$ and $r = 5$.

Worked example: area to circumference

$A = 16\pi\ \text{cm}^2$. Find $C$.

$r^2 = 16$, so $r = 4$ cm. $C = 2\pi r = 2 \times \pi \times 4 = 8\pi$ cm

Partial and composite figures

Semicircle: area $= \tfrac{1}{2}\pi r^2$. Perimeter $= \tfrac{1}{2}(2\pi r) + d = \pi r + d$. Do not forget the straight edge.
Quarter circle: area $= \tfrac{1}{4}\pi r^2$. Perimeter $= \tfrac{1}{4}(2\pi r) + 2r$.
Ring (annulus): outer area minus inner area $= \pi R^2 - \pi r^2$.
Shaded region: find the area of the big shape, find the area of the hole, subtract. A circle inside a square: $\text{side}^2 - \pi r^2$.
Rectangle with rounded ends (a track): two straight sides plus one full circle made from the two semicircles.

Worked example: shaded corners

A circle of radius 3 fits inside a $6 \times 6$ square. Shaded area $= 36 - 9\pi$ square units.

Read the figure twice. Ask: is this a whole circle, a half, or a quarter? Is the given length a radius or a diameter? Is the question about the edge (circumference, a length) or the inside (area, square units)?

25 problems. Show your work in the space under each problem, then put a box around your answer. Leave every answer exact, in terms of $\pi$ (for example $8\pi$ cm). Do not substitute 3.14. On screen, the eye at the right of each problem shows or hides the worked solution.