ACT and SAT Math Tips: Squares and Circles

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If you see a square inscribed in a Circled Squarescircle on the ACT or  SAT, the test makers are assessing your knowledge of 45:45:90 triangles.

What? Huh? The question is about squares and circles, not triangles! Listen closely, my ACT and SAT cohorts: Hidden triangles are often the key to solving the most difficult geometry questions.

That's right. Triangles are the most prominent figure on college admissions tests, mainly because they have several tricky properties. Learn those properties and how to spot hidden triangles and most geometry questions will become welcome puzzles rather than dreaded test questions. Let’s look at an example of an inscribed square problem:

Begin by labeling the diagram with the information provided in the question. Notice that the diameter of the circle is also the diagonal of the square:

If you can find the length of the diagonal, you can find the diameter and radius. Use your knowledge of triangles to find this information:

The two triangles are 45:45:90 triangles, so the hypotenuse is the length of the side multiplied by the square root of 2. The radius is one-half of the diameter:

Now that you know the radius of the circle, you can solve for the area:








The correct answer is (C).

Always be on the lookout for hidden triangles on SAT geometry questions. If you are graced with a question that has a square inscribed in a circle, know that the diagonal of the square, found using 45:45:90 triangle properties, leads to the discovery of the radius.

PowerScore Practice Prep

Can you solve the following math questions? The answers are listed below.
  1. A square is inscribed in a circle. The area of the circle is 50π. What is the perimeter of the square?
  2. A square is inscribed in a circle. If the area of the square is 36, what is the circumference of the circle?
  3. A square is inscribed in a circle. If the diameter of the circle is 4, what is the area of the square?


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Photo: Circled Squares, courtesy of zeevveez

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