AMC 10 · 2012 · #2
Grade 7 geometry-2dA circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?
Pick an answer.
AMC 10 2012 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 of radius $5$ is inscribed in a rectangle whose length-to-width ratio is $2:1$. Find the area of the rectangle.
Givens: The circle has radius $5$; The circle is inscribed in the rectangle, so it touches all four sides; The length of the rectangle is twice its width (ratio $2:1$); The figure shows the circle centered inside the rectangle, spanning it top-to-bottom
Unknowns: The area of the rectangle
Understand
Restated: A circle of radius $5$ is inscribed in a rectangle whose length-to-width ratio is $2:1$. Find the area of the rectangle.
Givens: The circle has radius $5$; The circle is inscribed in the rectangle, so it touches all four sides; The length of the rectangle is twice its width (ratio $2:1$); The figure shows the circle centered inside the rectangle, spanning it top-to-bottom
Plan
Primary tool: #1 Draw a Diagram
Secondary: #7 Identify Subproblems
The picture is the key. Tool #1 (Draw a Diagram) turns the word "inscribed" into a measurable fact: the circle is tangent to the top and bottom edges, so the vertical gap it spans — its diameter — is exactly the width of the rectangle. Once the width is pinned down, Tool #7 (Identify Subproblems) finishes in two small steps: get the length from the $2:1$ ratio, then multiply length by width for the area.
Execute — Answer: E
7.G.B.4 Step 1 Read the width off the circle
- The circle is inscribed, so it just touches the top edge and the bottom edge of the rectangle.
- The straight distance from the point where it touches the bottom to the point where it touches the top passes through the center — that is the circle's diameter.
- So the width of the rectangle equals the diameter, which is twice the radius: $2 \times 5 = 10$.
💡 A circle squeezed between two parallel walls fills exactly one diameter of space, so that diameter is the gap between the walls.
6.RP.A.1 Step 2 Use the ratio to get the length
- The length-to-width ratio is $2:1$, which means the length is $2$ times the width.
- The width is $10$, so the length is $2 \times 10 = 20$.
- This also matches the figure, where the rectangle is drawn twice as wide across as it is tall.
💡 A $2:1$ ratio just says the first amount is double the second, so scale the known side up by $2$.
4.MD.A.3 Step 3 Multiply length by width
- The area of a rectangle is length times width.
- With length $20$ and width $10$, the area is $20 \times 10 = 200$.
- That matches choice (E).
💡 Area of a rectangle counts the unit squares inside, and that is always length times width.
7.G.B.4 The circle is inscribed, so it just touches the top edge and the bottom edge of 6.RP.A.1 The length-to-width ratio is $2:1$, which means the length is $2$ times the widt 4.MD.A.3 The area of a rectangle is length times width. With length $20$ and width $10$, Review
Reasonableness: Check the two sides against the drawing: the width $10$ equals the diameter $2\times 5$, and the length $20$ is twice that — exactly the $2:1$ box in the figure. Area $200$ is the only choice that fits. The tempting wrong answer is (B) $100$, which comes from using the radius $5$ as the width instead of the diameter $10$; but a radius only reaches halfway across the circle, so the full width must be the diameter.
Alternative: Read the coordinates straight from the figure instead. The rectangle has corners $(0,0)$ and $(20,10)$, so it is $20$ wide and $10$ tall, giving area $20 \times 10 = 200$. The circle centered at $(10,5)$ with radius $5$ reaches from $y=0$ to $y=10$, confirming the diameter equals the height.
CCSS standards used (min grade 7)
7.G.B.4Know the formulas for area and circumference of a circle (Using the radius-to-diameter relationship ($\text{diameter} = 2 \times \text{radius}$) to see the inscribed circle spans a width of $10$.)6.RP.A.1Understand the concept of a ratio and use ratio language (Reading the $2:1$ length-to-width ratio as "the length is twice the width" to get length $20$.)4.MD.A.3Apply area and perimeter formulas for rectangles in real-world problems (Computing the rectangle's area as length times width, $20 \times 10 = 200$.)
⭐ An inscribed circle stretches exactly one diameter across, so its diameter is the box's short side — find that first, then the rest is just ratio and area.
⭐ An inscribed circle stretches exactly one diameter across, so its diameter is the box's short side — find that first, then the rest is just ratio and area.
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