Question 200·Medium·Geometry and Trigonometry

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A circle has circumference $18\pi$ centimeters. What is the area, in square centimeters, of the circle?

A given circumference measures the distance around a circle, while area measures the space inside it. Use $C=2\pi r$ to find the radius, the distance from the center to the edge, then put that radius into $A=\pi r^2$ . Don’t treat the diameter as the radius: squaring that longer length makes the area four times too large.

Hints

Hint 1
The radius runs from a circle’s center to its edge. Circumference uses $C=2\pi r$ , so what equation do you get when you replace $C$ with the given circumference?
Hint 2
The area formula is $A=\pi r^2$ . The radius gets squared before you multiply by $\pi$ . Which length have you found that can go in for $r$ ?

Step-by-step

Find the radius, then the area
Step 1
Turn the circumference into an equation
The circumference is the distance around the circle, and $r$ is the radius from the center to the edge. Since $C=2\pi r$ and the given circumference is $18\pi$ , replace $C$ with $18\pi$ :
$18\pi=2\pi r.$
Step 2
Find the radius
Divide both sides by $2\pi$ to get the radius $r$ alone:
$r=\frac{18\pi}{2\pi}.$
Cancel $\pi$ from the top and bottom:
$r=\frac{18}{2}.$
Divide:
$r=9.$
Step 3
Substitute the radius into the area formula
The area formula uses the radius squared, so put $r=9$ into $A=\pi r^2$ :
$A=\pi(9)^2.$
Step 4
Use the radius to find the area
Type $9^2$ into Desmos; it shows $81$ . Replace the square with $81$ :
$A=81\pi.$
Find the radius before you square it. The area is $81\pi$ square centimeters. Choice B.

Lessons that teach this

SAT Geometry and TrigonometryBeginnerCoreFind perimeter and area of plane figures

Question 200·Medium·Geometry and Trigonometry

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Strategy

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