AMC 10 · 2004 · #12

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference

tangent-circlespythagorean-theoremarea-difference identify-subproblemsconvert-to-algebra ↑ Prerequisites: pythagorean-theorem

📏 Short solution 💡 2 insights 📊 Diagram

Problem

An annulus is the region between two circles that share the same center O. The larger circle has radius b and the smaller one radius c, with b greater than c. OX is a radius of the larger circle, and the segment XZ is tangent to the smaller circle at Z. OY is the radius of the larger circle that passes through Z, and the labeled lengths are a = XZ, d = YZ, e = XY. What is the area of the annulus, written using only a, b, c, d, or e?

Pick an answer.

(A)

$\pi a^2$

(B)

$\pi b^2$

(C)

$\pi c^2$

(D)

$\pi d^2$

(E)

$\pi e^2$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The ring's area is the big circle minus the small circle, so it depends on b squared minus c squared. Reading the figure shows a right triangle hiding in the tangent line, and that right triangle is exactly what turns b squared minus c squared into a single labeled length. Draw the triangle, then let algebra finish.

1STEP 1

Ring equals big minus small

Cut the small disk out of the big one: the ring is π b² minus π c², that is π(b² - c²).

Area = π b² - π c² = π(b² - c²)

2STEP 2

Find the right triangle

Tangency at Z makes OZ meet XZ at a right angle, so OZX has legs c and a with hypotenuse b: b² = c² + a².

OX² = OZ² + XZ² → b² = c² + a²

3STEP 3

Swap in a single length

Rearranged, b² - c² is just a², so the ring's area collapses to π a² — choice (A).

b² - c² = a² → Area = π(b² - c²) = π a²

Answer

π a²

The answer pi a squared is the area of a circle with radius a, and it uses only labeled lengths, as required. It correctly ignores d and e, which never entered the ring's area. A quick sanity check: if the inner circle shrinks toward the center (c toward 0), the tangent length a grows toward b, and pi a squared grows toward pi b squared, the full big circle, exactly as a vanishing hole should give.

💡Key takeaway

A tangent line makes a right angle with the radius, so the Pythagorean theorem turns the ring's area into a single circle of radius a.

Ring equals big minus small
Find the right triangle
Swap in a single length

Ring equals big minus small

Find the right triangle

Swap in a single length

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