Geometry: Inscribed Squares Hide Right Triangles

A square that fits snugly inside a circle is inscribed in the circle. The square’s corners will touch, but not intersect, the circle’s boundary, and the square’s diagonal will equal the circle’s diameter. Also, as is true of any square’s diagonal, it will equal the hypotenuse of a 45°-45°-90° triangle. GRE questions about squares inscribed in circles are really questions about the hypotenuse of this hidden right triangle.

Here’s an example of an inscribed square problem.

To find the circle’s area, you’ll need to find the radius, which is half the diameter. Sketch the figure described in the question, and mark the diagonal of the square and, with that, the diameter of the circle.

The diameter/diagonal splits the inscribed square in to two right triangles that sit hypotenuse-to-hypotenuse. Both triangles have legs of 4 (since the square has sides of 4) and interior angles of 45°, 45°, and 90°. For either one, you can find the hypotenuse using the ratio of the triangle’s sides.

45°-45°-90° Triangle Ratio
leg : leg : hypotenuse = s : s : s√2

• s = 4
• ⇒ s√2 = 4√2

Now, halve the triangle’s hypotenuse to find the radius of the circle.

• d = s√2
• ⇒ d = 4√2
• r = d2
• ⇒ r = 2√2

Finally, plug the circle’s radius in to the area formula.

Area of a Circle = πr²

• π(2√2)²
• π(2²)(√2²)
• π(4)(2)
• 8π
• Thus, (C) is correct.

Look out for hidden triangles in SAT geometry questions. If you get a question with a square inscribed in a circle, remember that the diagonal of the square doubles as the hypotenuse of a a 45°-45°-90° triangle. Finding that hypotenuse will likely be the key to answering the question.

Want more math tips like these? Check out the GRE Quantitative Reasoning Bible! In the meantime, try a few more practice problems.

Answers: 1. D 2. D 3. C

Photo: “Squares, circles and lines. Oh My!“, courtesy of Kenny Louie