Perimeters and areas of plane figures (circles)

Advanced Math 7 · Unit 12 · circles-arcs-sectors · Teacher edition

circles-arcs-sectors

Name ______________________________Date ______________

1.
A circle has diameter $14$ cm. Its circumference is $k\pi$ cm. Enter $k$ .
Answer: ______________
2.
A sector of a circle has radius $9$ cm and central angle $120^\circ$ . Its area is $k\pi$ cm². Enter $k$ .
Answer: ______________
3.
A square has side $8$ cm. A quarter circle with its center at one corner of the square and radius $8$ cm is cut away. Using $\pi \approx 3.14$ , what is the area of the remaining piece, in cm²?
Answer: ______________
4.
A sector has radius $6$ cm and central angle $60^\circ$ . What is its area?
1. (A)
  $\dfrac{3}{2}\pi$ cm²
2. (B)
  $2\pi$ cm²
3. (C)
  $6\pi$ cm²
4. (D)
  $36\pi$ cm²
5.
A circle is inscribed in a square of side $12$ , touching all four sides. The total area of the four corner regions — inside the square but outside the circle — can be written as $a - b\pi$ for positive integers $a$ and $b$ . What is $a + b$ ?
1. (A)
  $108$
2. (B)
  $144$
3. (C)
  $156$
4. (D)
  $180$
5. (E)
  $288$

Answer key — Perimeters and areas of plane figures (circles)

1.
14circles-arcs-sectors-01
$C = \pi d = 14\pi$ , so $k = \mathbf{14}$ . ( $28$ treats $14$ as the radius.)
2.
27circles-arcs-sectors-02
$A = \frac{120}{360} \times \pi \times 81 = 27\pi$ , so $k = \mathbf{27}$ . (The arc length would be $\frac{1}{3} \times 18\pi = 6\pi$ — a different question.)
3.
13.76circles-arcs-sectors-03
Quarter circle $= \frac{1}{4}\pi(8^2) = 16\pi \approx 50.24$ . Remaining $= 64 - 50.24 = \mathbf{13.76}$ cm².
4.
(C)
$6\pi$ cm²
circles-arcs-sectors-04
$A = \frac{1}{6} \times 36\pi = \mathbf{6\pi}$ cm². ( $2\pi$ is $\frac{1}{6} \times 2\pi r$ — the arc length, not the area; $\frac{3}{2}\pi$ uses $r = 3$ , half the radius; $36\pi$ is the whole circle.)
5.
(D)
$180$
circles-arcs-sectors-05
$r = 6$ , so the corners have area $144 - 36\pi$ and $a + b = 144 + 36 = \mathbf{180}$ . ( $288$ uses $r = 12$ , the diameter; $156$ uses $12\pi$ for the circle; $144$ forgets the circle.)