In this series, we will cover many types of geometric scenarios encountered on the SAT and ACT.  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 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 question 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.

For circles:

• d=2r and all lines from the center to the exterior equal r.
• C = 2πr = πd
• A = πr²
• 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.
• The diagonal equals (length of a side * √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.  (Inscribed means that the circle fits perfectly inside the square with its edges touching the side of the square like in the diagram)

Test your SAT math skills with this SAT multiple choice practice question.

For squares:

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).

Shaded Areas

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 because it involves a circular shape and a square/rectangle.

The picture above depicts a circle perfectly inscribed in a square.  What is the area of the shaded region?

By knowing the area of the large square we also know the lengths of its sides. (Note that 64 is a perfect square, which should be a clue.) If the side is 8, then so is the diameter, which means the radius equals 4. In the image, we can see that the “larger figure” is the top-right square bordered by two radii and the outer border. How do we know that it’s a square? Two pieces of information: All sides are equal to 4, and the radius meets the large square at a right angle because it is a tangent. In this instance, the area of the smaller square equals 16. Since the interior angle is 90-degrees (360/4), the area of the sector of the circle can be represented by A = πr²/4. So that A = 16π/4 = 4π.

A(shaded) = A(small square) – A(sector) = 16 – 4π

Make sure to check out Geometry Series Part 2 and Geometry Series Part 3!