In The Figure Above, If The Square Inscribed In The Circle Has An Area GMAT Problem Solving

Question:

In the figure above, if the square inscribed in the circle has an area of 16, what is the area of the shaded region?

A) 2π−1
B) 2π−4
C) 4π−2
D) 4π−4
E) 8π−4

“In the figure above, if the square inscribed in the circle has an area''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT quant section is concerned with the logical thinking capacity of the candidates and their skills in solving quantitative problems. The candidates must select the right option by calculating the entire sum properly by maintaining mathematical rules. The students must hold basic ideas of mathematical calculations to solve GMAT Problem Solving questions. The GMAT Quant topic in the problem-solving part represents mathematical problems that can be solved with better calculative skills. The candidates can practice questions by answering from the book “The Official Guide for GMAT Review 2017”.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:

Given:

• the area of the square inscribed in the circle is 16

Find Out:

• the area of the shaded region

Since the square is inscribed in a circle, then as per the rule, we can say,
Diameter of the circle = diagonal of the square.

Let the side of a square is a units.
According to the formula of the length of the diagonal of a square, we can say,
The diagonal of the square = a√2 units.

The area of the square as given in the question is 16 units
Therefore, the side of the square = 4 units (since the side of a square= √area )
Then a= 4 units
Therefore, the diagonal of the square = 4√2 units = diameter of the circle.

Since both the circle and square hold a high extent of symmetry, we can say that,
Area of shaded region = (Area of the Circle – Area of the square)/4

Diameter of circle = 4√2, therefore radius= 4√2/2 = 2√2
Area of circle = π(radius)^2 = π(2√2)^2 = 8π.

Given, the area of the square is 16.
Therefore, area of shaded region = 8π–16/4 = 2π – 4

Correct Answer: (B)

Approach Solution 2:

The problem statement suggests that:

Given:

• the area of the square inscribed in the circle is 16

Find Out:

• the area of the shaded region

Since the area of the square is 16, the sides of the square are 4 units.
As we know that if a square is inscribed in a circle, the diagonal of the square is the diameter of the circle.
If a diagonal is drawn in a square, the square is divided into two 45-45-90 triangles.
The sides of a 45-45-90 triangle are in a ratio of 1:1:√2
The side of the square= 4 units
Therefore, the diagonal of the square = 4√2 (since diagonal= side√2)
Hence, the diameter of the circle =4√2.
Therefore, the radius of the circle= 2√2, (since the radius is half of the diameter)
Therefore, the area of the circle is = πr^2 = π(2√2)^2 = 8π.

If we subtract the area of the square from the area of the circle, we get the total area outside of the square, but inside of the circle= 8π - 16
The area of those portions outside the square and inside the circle consists of four equal parts.
The shaded area is only one of those parts.
Therefore, the area of the shaded region is 1/4 * (8π - 16) = 2π - 4.

Correct Answer: (B)

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