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Problem 550

# A circle is inscribed in an equilateral triangle, whose side is 12\. Find, to the nearest integer, the difference between the area of the triangle and the area of the circle. (Use $$\pi=3.14$$ and $$\sqrt{3}=1.73 .$$ )

Expert verified
The area of the equilateral triangle is approximately 72.03, and the area of the inscribed circle is approximately 37.81. The difference between the areas is about 34.22, which, when rounded to the nearest integer, is 34.
See the step by step solution

## Step 1: Find the area of the equilateral triangle

Using the formula for the area of the equilateral triangle, we have: Area = $$\frac{\sqrt{3}}{4} \times 12^2 = \frac{\sqrt{3}}{4} \times 144$$ Since $$\sqrt{3} = 1.73$$, we can substitute the value: Area = $$\frac{1.73}{4} \times 144 \approx 72.03$$

## Step 2: Find the radius of the inscribed circle

Using the formula for the radius of the inscribed circle, we have: Radius = $$\frac{12}{2\sqrt{3}} = \frac{12}{2 \times 1.73}$$ Radius = $$3.47$$, rounding to two decimal places

## Step 3: Find the area of the inscribed circle

Using the formula for the area of a circle, we have: Area = $$\pi \times (3.47)^2$$ Since $$\pi = 3.14$$, we can substitute the value: Area = $$3.14 \times 12.04 \approx 37.81$$

## Step 4: Find the difference between the areas and round to the nearest integer

Now that we have both areas, we can calculate the difference: Difference = Area of triangle - Area of circle Difference = $$72.03 - 37.81 \approx 34.22$$ To round to the nearest integer, the difference is: Difference = $$34$$.

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