The competitive exams loves to incorporate more than one familiar shape in a diagram. The most popular combination is of a circle and a square.
Squares can be inscribed in circles, and circles can be inscribed in square. A circle inscribed in a square is a little easier to work with, so let’s start there.
The word “inscribed” has a very particular meaning. To say that one figure is “inscribed” in another doesn’t mean that it is simply “inside” that other figure. It is as big as possible, sharing some of the same perimeter.
Circle in a Square
When a circle is inscribed in a square, the top of the circle touches the top border of the square, the rightmost point of the circle touches the right border of the square, and so on.
Thus, these two figures have some measurements in common. The diameter of the circle is equal to the length of one side of the square. You’ll see this most easily if you draw a diameter line across the circle from left to right, or from top to bottom.
All other relationships flow from the equality of the diameter and side length. From diameter, you can find radius, circumference, perimeter, and area. From side length, you can find perimeter and area.
Square in a Circle
The other variation is when the square is the smaller of the two figures. In this case, all four vertices of the square are tangent (touch) the perimeter of the circle.
Again, the diameter of the circle is key. But now, the diameter of the circle is not equal to a side length of the square. This time, it’s most helpful to draw the diameter diagonally, linking two opposite vertices of the square.
The diameter of this circle is equal to the “diagonal” of the square. The diagonal has a consistent relationship with the side length of a square, though it’s a little harder to work out.
You may be more familiar with the characteristics of an equilateral right triangle, otherwise known as a 45-45-90 right triangle. When you draw the diagonal of a square, you’ve just created two 45-45-90 triangles.
So, use the side length rules for a 45-45-90 triangle. If the side of the square is x, the diagonal is x times rt(2).
Thus, if the side of the square is x, the diameter of the circle is x times rt(2). The numbers can get a little messy, but again, the relationships are consistent.
In Data Sufficiency questions, you may not have to work out any algebra. You may only need to determine what could be worked out.
If that is the case, remember that any time a circle and a square are related by inscription, the relationships are known. From any one piece of information about one of the two figures, you can derive any common piece of information about either figure.
These relationships even extend beyond the inscription of one figure in another. Let’s say you have a circle inscribed in a square, which is itself inscribed in a circle. Once you are given a characteristic of the inner circle, you can find any measurement of the square. And once you know the measurements of the square, you can find any measurement of the outer circle!
A three-figure diagram like the one I’ve just described would probably be considered a difficult question, but once you understand these relationships, it doesn’t have to be difficult for you.
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