Geometry Series Part 1: Circles Inscribed in Squares

In this series, we will cover many types of geometric scenarios encountered on the GRE. A basic knowledge of simple formulas (area, perimeter, etc.) is essential, but there are numerous shortcuts to geometry questions that will save you time. Today, we’ll explore circles inscribed in squares.

Some Things to Remember

• The center of the square is the same point as the center of the circle
• Draw lines! Depending on what the stimulus asks for, draw in lines that create simple shapes. (Squares can be turned into triangles, for example.)
• Shared angles will normally not be explicitly stated, unless necessary.
• Trust the pictures, but not too much. Inferences must be drawn from fact. Just because it looks like 90-degrees doesn’t mean it is! (Many of these common inferences will be detailed in this series.)
• Lengths cannot be negative. Be careful in DS questions that pose equations in the context of quadratic equations with two solutions. If one solution is negative and the other is positive, only the positive solution remains and the information is sufficient.

For circles:

• d=2r and all lines from the center to the exterior equal r.
• C = 2πr = πd
• A = πr²
• NEVER use 2πr² unless you are adding the areas of identical circles!
• Tangent lines create right angles with the radius that meets that tangent.
• If you know r, you know everything about the circle!
• Use π = 22/7 with caution. Remember 22/7 > π.

For squares:

• The diagonal equals s√2, since it creates 45-degree angles.
• The intersection of the diagonals creates a right angle.
• When a circle is inscribed inside a square, the side equals the diameter.

Usually, you will be provided with one bit of information that tells you a whole lot, if not everything. If given the length of the side of the square in the above image, we can actually find the length of the hypotenuse of the internal triangle (s = d = 2r, so the hypotenuse = (s√2)/2).

Find the large area and subtract the small area from it. When dealing with circles along with other figures, eliminate answer choices that ONLY have π’s in them or don’t have any at all. Typically, your answer will look like x + yπ.

Two important takeaways:

1. Never assume without proof.