AMC 10 · 2010 · #5
Grade 7 geometry-2dThe area of a circle whose circumference is 24π is kπ. What is the value of k?
Pick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A circle has circumference $24\pi$. Its area can be written as $k\pi$. Find the number $k$.
Givens: The circle's circumference is $24\pi$.; The circle's area is written in the form $k\pi$.; Answer choices: (A) $6$, (B) $12$, (C) $24$, (D) $36$, (E) $144$.
Unknowns: The value of $k$, which is the area divided by $\pi$.
Understand
Restated: A circle has circumference $24\pi$. Its area can be written as $k\pi$. Find the number $k$.
Givens: The circle's circumference is $24\pi$.; The circle's area is written in the form $k\pi$.; Answer choices: (A) $6$, (B) $12$, (C) $24$, (D) $36$, (E) $144$.
Plan
Primary tool: #4 Introduce a Variable
Secondary: #11 Work Backwards, #7 Identify Subproblems
The circumference and the area of a circle are not directly linked, but both are built from the same radius $r$. So name that radius with a letter (Tool #4, Introduce a Variable). The given circumference is the finished output of the formula $C = 2\pi r$; run that formula in reverse to recover $r$ (Tool #11, Work Backwards). That splits the task into two clean subproblems (Tool #7): first find $r$ from the circumference, then feed $r$ into the area formula. Solving in this order means no guessing among the five choices.
Execute — Answer: E
7.G.B.4 Step 1 Name the radius
- Let $r$ be the radius of the circle.
- The circumference formula says $C = 2\pi r$.
- The problem gives $C = 24\pi$, so set the formula equal to the given value.
💡 Every circle fact rides on its radius, so pin the radius down with a letter first.
6.EE.B.7 Step 2 Work back to the radius
- Undo the formula to get $r$ by itself.
- Divide both sides of $2\pi r = 24\pi$ by $2\pi$.
- The $\pi$ cancels and $24 \div 2 = 12$, so $r = 12$.
💡 Dividing by the same factor on both sides peels the radius out of the circumference formula.
7.G.B.4 Step 3 Compute the area
- Now use the area formula $A = \pi r^2$ with $r = 12$.
- Squaring gives $12^2 = 144$, so $A = 144\pi$.
- Since the area is written as $k\pi$, matching the two forms gives $k = 144$.
- The answer is $\textbf{(E)}\ 144$.
💡 Once the radius is known, the area is a single plug-in, and squaring $12$ makes the number jump.
7.G.B.4 Let $r$ be the radius of the circle. The circumference formula says $C = 2\pi r$ 6.EE.B.7 Undo the formula to get $r$ by itself. Divide both sides of $2\pi r = 24\pi$ by 7.G.B.4 Now use the area formula $A = \pi r^2$ with $r = 12$. Squaring gives $12^2 = 144 Review
Reasonableness: Check the radius against the circumference: $2\pi (12) = 24\pi$, which matches the given circumference exactly. A radius of $12$ is fairly large, so an area of $144\pi$ (much bigger than the circumference number $24$) makes sense because area grows with the square of the radius. Choice (E) $144$ is the only option equal to $12^2$, which is a strong sign it is right.
Alternative: Relate area to circumference without finding $r$ explicitly. From $C = 2\pi r$ we get $r = \dfrac{C}{2\pi}$, so $A = \pi r^2 = \pi\left(\dfrac{C}{2\pi}\right)^2 = \dfrac{C^2}{4\pi}$. Plugging $C = 24\pi$ gives $A = \dfrac{(24\pi)^2}{4\pi} = \dfrac{576\pi^2}{4\pi} = 144\pi$, so $k = 144$ again.
CCSS standards used (min grade 7)
7.G.B.4Know the formulas for area and circumference of a circle (Setting the circumference formula $2\pi r$ equal to $24\pi$ and applying the area formula $\pi r^2$ once the radius is known.)6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Solving $2\pi r = 24\pi$ by dividing both sides by $2\pi$ to get $r = 12$.)
⭐ Both the circumference and the area come from the radius, so back the radius out of the circumference first, then square it in the area formula to get $k = 144$.
⭐ Both the circumference and the area come from the radius, so back the radius out of the circumference first, then square it in the area formula to get $k = 144$.
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