Worked Examples: GRE Geometry

Example 1: Triangle Angles in a Ratio (Medium)

The three interior angles of a triangle are in the ratio $2 : 3 : 4$. What is the measure, in degrees, of the largest angle?

Solution. Let the angles be $2k$, $3k$, $4k$. Their sum is $180°$: $2k + 3k + 4k = 9k = 180 \;\Rightarrow\; k = 20.$ The largest angle is $4k = 4(20) = 80°$.

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Example 2: Circle — Area to Circumference (Medium)

A circle has area $36\pi$. What is its circumference?

Solution. From $A = \pi r^2 = 36\pi$, we get $r^2 = 36$, so $r = 6$. Then $C = 2\pi r = 2\pi(6) = 12\pi.$ The trap answer $36\pi$ simply repeats the area; the two quantities are numerically different because area depends on $r^2$ while circumference depends on $r$.

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Example 3: Hidden Right Triangle in a Rectangle (Medium–Hard)

A rectangle has a length of 12 and a diagonal of 13. What is the area of the rectangle? (Figure not drawn to scale.)

Solution. The diagonal, length, and width form a right triangle: $\text{width} = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5$. Area $= 12 \times 5 = 60$.

The trap answer 156 comes from multiplying length by diagonal ($12 \times 13$) — the diagonal is the hypotenuse, not a side.

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Example 4: Cylinder Volume (Medium)

A right circular cylinder has radius 3 and height 10. What is its volume?

Solution. $V = \pi r^2 h = \pi (3)^2 (10) = \pi (9)(10) = 90\pi.$

A frequent error is $V = \pi r h = 30\pi$, forgetting to square the radius; another is $\pi(3)^3 = 27\pi$, confusing the cylinder with a cube-like power.

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Example 5: Parallel Lines and a Transversal (Medium)

In a figure (not drawn to scale), lines $\ell$ and $m$ are parallel and are cut by a transversal. One interior angle on a given side of the transversal measures $65°$. What is the measure of the other interior angle on the same side?

Solution. Same-side (co-interior) interior angles between parallel lines are supplementary: $65° + x = 180° \;\Rightarrow\; x = 115°.$ Do not assume the two angles are equal from the picture — only alternate interior and corresponding angles are equal; same-side interior angles are supplementary.

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Example 6: Regular Polygon Interior Angle (Medium)

What is the measure of each interior angle of a regular hexagon?

Solution. The interior angles of an $n$-gon sum to $(n - 2) \times 180°$. For $n = 6$: $\frac{(6 - 2) \times 180°}{6} = \frac{720°}{6} = 120°.$ The trap answer $108°$ is the interior angle of a regular pentagon ($n = 5$).