AMC 10 · 2002 · #7
Grade 7 geometry-2dA 45∘ arc of circle A is equal in length to a 30∘ arc of circle B. What is the ratio of circle A's area and circle B's area?
Pick an answer.
AMC 10 2002 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 $45^\circ$ arc cut from circle A has the same length as a $30^\circ$ arc cut from circle B. Using only that fact, find the ratio of circle A's area to circle B's area.
Givens: The length of a $45^\circ$ arc of circle A equals the length of a $30^\circ$ arc of circle B; An arc's length is the same fraction of its circle's full circumference as its angle is of $360^\circ$; Answer choices: (A) $4/9$, (B) $2/3$, (C) $5/6$, (D) $3/2$, (E) $9/4$
Unknowns: The ratio $\dfrac{\text{area of circle A}}{\text{area of circle B}}$
Understand
Restated: A $45^\circ$ arc cut from circle A has the same length as a $30^\circ$ arc cut from circle B. Using only that fact, find the ratio of circle A's area to circle B's area.
Givens: The length of a $45^\circ$ arc of circle A equals the length of a $30^\circ$ arc of circle B; An arc's length is the same fraction of its circle's full circumference as its angle is of $360^\circ$; Answer choices: (A) $4/9$, (B) $2/3$, (C) $5/6$, (D) $3/2$, (E) $9/4$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
No lengths are given, so name what is missing: let $r_A$ and $r_B$ be the two radii (tool #4). Once each arc is written in terms of its radius, the 'equal length' fact becomes a single equation linking $r_A$ and $r_B$. Tool #7 (Identify Subproblems) splits the work into two clean stages: first find the ratio of the radii, then turn that into the ratio of the areas — because area depends on the radius squared, these are genuinely different questions. Tool #3 (Eliminate Possibilities) guards the finish: the larger-angle arc belongs to the smaller circle, so circle A is smaller and the area ratio must be less than $1$, which already throws out (D) $3/2$ and (E) $9/4$.
Execute — Answer: A
7.G.B.4 Step 1 Write each arc's length
- An arc is a slice of the full circumference $2\pi r$, and its share is its angle out of $360^\circ$.
- Circle A's $45^\circ$ arc is $\frac{45}{360}=\frac{1}{8}$ of its circumference, giving length $\frac{1}{8}\cdot 2\pi r_A=\frac{\pi r_A}{4}$.
- Circle B's $30^\circ$ arc is $\frac{30}{360}=\frac{1}{12}$ of its circumference, giving length $\frac{1}{12}\cdot 2\pi r_B=\frac{\pi r_B}{6}$.
💡 An arc is just a fraction of the way around the circle, so its length is that same fraction of the whole circumference.
6.RP.A.3 Step 2 Set the arcs equal, find the radius ratio
- The two arcs have equal length, so $\frac{\pi r_A}{4}=\frac{\pi r_B}{6}$.
- The common factor $\pi$ cancels from both sides, leaving $\frac{r_A}{4}=\frac{r_B}{6}$.
- Cross-multiplying gives $6r_A=4r_B$, so $\frac{r_A}{r_B}=\frac{4}{6}=\frac{2}{3}$.
- Circle A's radius is two-thirds of circle B's — A is the smaller circle, exactly as expected since its arc spanned the larger angle.
💡 Equal arcs from a wider and a narrower angle can only balance if the wider-angle circle is smaller, and the ratio makes that exact.
7.G.B.4 Step 3 Square the ratio to compare areas
- A circle's area is $\pi r^2$, so the ratio of two circles' areas is the ratio of their radii squared: $\frac{\pi r_A^2}{\pi r_B^2}=\left(\frac{r_A}{r_B}\right)^2$.
- Substituting $\frac{r_A}{r_B}=\frac{2}{3}$ gives $\left(\frac{2}{3}\right)^2=\frac{4}{9}$.
- The area ratio is less than $1$, confirming circle A is the smaller of the two.
- This matches choice (A).
💡 Because area grows with the square of the radius, scaling the radius by a factor scales the area by that factor squared.
7.G.B.4 An arc is a slice of the full circumference $2\pi r$, and its share is its angle 6.RP.A.3 The two arcs have equal length, so $\frac{\pi r_A}{4}=\frac{\pi r_B}{6}$. The co 7.G.B.4 A circle's area is $\pi r^2$, so the ratio of two circles' areas is the ratio of Review
Reasonableness: Check the direction and the size. The $45^\circ$ arc spans a bigger angle than the $30^\circ$ arc, yet both lengths match, so circle A must be smaller to make its wider slice come out the same length — and indeed $r_A/r_B=2/3<1$ and area ratio $4/9<1$, both saying A is smaller, which is consistent. The size is sensible too: squaring $2/3$ shrinks it from $0.67$ to about $0.44$, and $4/9\approx 0.44$ sits neatly between the smaller decoys, not near $1$. The trap answer $2/3$ is exactly the radius ratio left un-squared, and $9/4$ flips the fraction the wrong way; the area ratio for the smaller circle over the larger has to be below $1$.
Alternative: Reason with proportions and skip the algebra. Equal arc length means (angle share) times circumference is equal, so $\frac{1}{8}(2\pi r_A)=\frac{1}{12}(2\pi r_B)$; since $\frac{1}{8}$ is $\frac{2}{3}$ of $\frac{1}{12}\cdot ?$ — more simply, circle A uses $\frac{1}{8}$ of its circumference where B uses $\frac{1}{12}$, and $\frac{1}{8}:\frac{1}{12}=3:2$, so A's circumference (hence radius) is $\frac{2}{3}$ of B's. Squaring the radius ratio gives the area ratio $\frac{4}{9}$.
CCSS standards used (min grade 7)
7.G.B.4Know the formulas for area and circumference of a circle (Writing each arc as a fraction of its circumference $2\pi r$, and using area $=\pi r^2$ to turn the radius ratio into the area ratio.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Setting the two arc lengths equal, cancelling $\pi$, and cross-multiplying to get the radius ratio $r_A/r_B=2/3$.)
⭐ Equal arcs let you find the ratio of the radii; then square that ratio to get the ratio of the areas, because a circle's area grows with the radius squared.
⭐ Equal arcs let you find the ratio of the radii; then square that ratio to get the ratio of the areas, because a circle's area grows with the radius squared.
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