Question:

A circle is inscribed in an equilateral triangle of side 24 cm, as shown above. What is the area of the shaded region?

1. 148√3−36π cm^2
2. 144√3−36π cm^2
3. 144√3−48π cm^2
4. 121√3−48π cm^2
5. 121√3−36π cm^2

Correct Answer: (C)

Approach Solution : 1

Let us use the following strategy,
The Area of Shaded Region = [Equilateral triangle - Area of Circle]

Area of equilateral triangle = (√3/4)∗ (24^2) = 144√3
Area of Circle = π(r^2)

The ideas is that the, radius of circle inscribed in an equilateral triangle = (1/3) Height of equilateral = (1/3)*(√3/2)*Side
Therefore the radius of Circle = (1/3)∗(√3/2)∗24 = 4√3
Now, Area of Circle = π[(4√3)^2] = 48π
As a result, the shaded Area = [144√3−48π]

Approach Solution : 2

When determining the shaded area of the triangle, we take into account the relationship between the triangle's area, semi-perimeter, and inner circle's radius, which is R=A/s.
We are aware that the relationship between the area of a triangle, its semi-perimeter, and its radius is R=A/s, where A is the triangle's area, s is its semi-perimeter, and R is its radius.

It is given that an equilateral triangle has a side of 24 cm.
Note that semi perimeter s can be written as, s = (24+24+24) / 2 = 36cm.

The formula for the area A for the equilateral triangle is √3/2(side)^2 .
So, the area will be √3/2 * (24^2) = 1443cm^2 .

Now that we know what A and s are worth, we can quickly calculate what R is worth by using the formula below,
R= (144√3) / 36 = 4√3cm

Since we now have the radius, we can proceed further,
Formula for area for the circle is = π(radius)^2
Area of the circle will be π(R^2) = π*(4^2)*3 = 48π

Therefore the area of the shaded portion will be calculated as the area of the circle being subtracted from the total area of the triangle.
With the above description, the result is 144√3−48π cm^2

“A circle is inscribed in an equilateral triangle of side 24 cm” - is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

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