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. 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. Leave every answer exact, in terms of $\pi$ (for example $8\pi$ cm). Do not substitute 3.14. The eye at the right of the row shows or hides the worked solution.