12 multiple-choice questions, progressively harder.
A circle has exact circumference 27.6π27.6\pi27.6π cm. What is its exact area?
Solution
Correct answer: C
First recover the radius.
A disk has area 98π98\pi98π square centimeters. What is its exact circumference?
Correct answer: A
The radius is the positive square root after dividing the area by pi.
A square of side 6.56.56.5 cm has all four vertices on a circle. What is the disk's exact area?
Correct answer: D
The square diagonal is the circle diameter. Use Pythagoras, then halve that diagonal.
Two squares have the same positive side length. Circle I passes through the four vertices of its square. Circle II touches all four sides of its square from the inside. What is disk I's area divided by disk II's area?
Correct answer: B
If the common side is a, circle I has radius a divided by square root 2, while circle II has radius a/2.
A circle has radius 8.28.28.2 cm, and 3.14<π<3.153.14<\pi<3.153.14<π<3.15. The area estimate uses 3.143.143.14. Which positive number bounds the possible underestimate most tightly using these data?
The error in pi is less than 0.01, and area multiplies it by the radius squared.
For a radius-101010 circle, inscribed and circumscribed squares have areas 200200200 and 400400400 square units. What bounds on pi follow?
The disk area lies strictly between the two square areas.
Divide all three quantities by 100 to obtain the stated interval.
At every arc bisection, a polygon side's squared length becomes less than half its previous value. An initial side is 121212 cm long. After four bisections, which bound is guaranteed from this fact?
Apply the factor to the squared length, then take the positive square root.
A circle has radius 555 cm. A regular inscribed hexagon and a regular tangent hexagon with parallel sides bound it. Which pair gives their areas, inner first?
The inner hexagon has perimeter 303030 and apothem 53/25\sqrt3/253/2. The outer perimeter is 20320\sqrt3203 and its apothem is 555.
Two circles have circumferences 12π12\pi12π cm and 22π22\pi22π cm. What is the second disk's area divided by the first disk's area?
The circumference ratio is the radius ratio. Square it to compare areas.
The inner-polygon area is Ph/2Ph/2Ph/2. Which changes in PPP and hhh make it approach Cr/2Cr/2Cr/2 in the circle-area argument?
The inner perimeter approaches the circumference; the perpendicular center-to-side distance approaches the radius. Those are the two factors in the triangle-area sum.
A circle's radius increases by 15%15\%15%. Which pair gives the percentage increases in circumference and area?
The length factor is 1.15. Circumference uses that factor; area uses its square.
The new area is 132.25 percent of the old area, an increase of 32.25 percent.
Certified bounds give 31.232<C<31.23831.232<C<31.23831.232<C<31.238, in centimeters. Which value of C rounded to the nearest hundredth is forced by these bounds alone?
The interval crosses the rounding boundary 31.235. Values 31.234 and 31.236 both satisfy the bounds but round to 31.23 and 31.24 respectively. The enclosure must be tightened before one rounded answer is certified.
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