12 multiple-choice questions, progressively harder.
A sector in a circle of radius 999 cm has area 27π cm227\pi\text{ cm}^227π cm2. What is its central angle in degrees?
Solution
Correct answer: C
Compare its area with the full disk area to recover the fraction of a turn.
A sector has area 45 cm245\text{ cm}^245 cm2 and radius 666 cm. What is its central angle in radians?
Correct answer: A
Undo multiplication by half the radius squared.
A minor circular segment comes from a 60∘60^\circ60∘ sector of radius 888 cm. What is its area?
Correct answer: D
The radii and chord form an equilateral triangle. Its altitude is half its side times the square root of three.
Subtract the triangle area from the sector area.
A minor circular segment has central angle 90∘90^\circ90∘ in a radius-666 cm circle. What is its complete perimeter?
Correct answer: B
The boundary is the quarter-circle arc plus the chord. The chord is the hypotenuse of a right triangle with both legs 6 cm.
An annulus has outer radius 101010 cm and area 84π cm284\pi\text{ cm}^284π cm2. What is its inner radius?
Subtract the inner disk area from the outer disk area, then take a positive square root.
An annulus has area 96π cm296\pi\text{ cm}^296π cm2 and total boundary length 48π48\pi48π cm. What is its width?
The perimeter determines the sum of radii. Factor the area to recover their difference, which is the width.
A sector has radius 777 cm and angle 5π/75\pi/75π/7 radians. What is its complete perimeter?
Multiply radius by the radian angle to find the arc, then add both radii.
The upper half of an annulus has outer radius 999 cm and inner radius 333 cm. What is its complete perimeter, including both straight connectors?
The two semicircular arcs contribute the sum of their half-circumferences. Each connector has length 6 cm.
A sector has radius 888 cm and area 24π cm224\pi\text{ cm}^224π cm2. What is the length of its selected arc?
Use the area relationship with radius and arc length.
A circle has radius 777 cm. The smaller angle between radii to two chord endpoints is 90∘90^\circ90∘. What is the area of the major circular segment?
Find the minor segment first, then subtract it from the disk. The major segment includes the triangle between the radii.
Two sectors have equal areas. One has radius 666 cm and angle 100∘100^\circ100∘. The other has radius 101010 cm. What is its angle in degrees?
For equal areas, angle times squared radius is the same when both angles use degrees.
A 120∘120^\circ120∘ part of an annulus has outer radius 999 cm and inner radius 666 cm. Which pair gives its area and complete perimeter?
Area 27π cm227\pi\text{ cm}^227π cm2.
Perimeter 6+10π cm6+10\pi\text{ cm}6+10π cm.
Area 15π cm215\pi\text{ cm}^215π cm2.
Perimeter 10π cm10\pi\text{ cm}10π cm.
Take one third of the difference of the disk areas. For the perimeter, the two arcs have combined length s and the two connectors have combined length L.
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