Find the circumference of a circle inscribed in an equilateral triangle if the height of the triangle is 14 in.
A)21.99 in.
B)87.96 in.
C)29.32 in.
D)43.98 in.

Question

Find the circumference of a circle inscribed in an equilateral triangle if the height of the triangle is 14 in. 
A)21.99 in. 
B)87.96 in. 
C)29.32 in. 
D)43.98 in.

Quizplus · Accepted Answer

In an equilateral triangle, the height intersects the base and the base is split into two congruent parts, each with a length equal to half of the base. Let's call the length of the base b.

Using the Pythagorean theorem, we can find the length of one of the sides of the equilateral triangle:

$(\frac{b}{2})^2 + 14^2 = b^2$
Simplifying, we get: 
$\frac{b^2}{4} + 196 = b^2$
$3b^2/4 = 196$
$b^2 = 256$
$b = 16$

So, the side length of the equilateral triangle is 16 inches.

The radius of the inscribed circle can be found by drawing perpendiculars from the center of the circle to each side of the equilateral triangle. This creates three smaller equilateral triangles within the big triangle, each with height equal to the radius of the inscribed circle.

Using the Pythagorean theorem again, we can find the radius:

$(\frac{16}{2})^2 + r^2 = r^2$
Simplifying, we get: 
$64/4 = r^2$
$r^2 = 16$
$r = 4$

So, the radius of the inscribed circle is 4 inches.

Finally, we can use the formula for the circumference of a circle to find the circumference of the inscribed circle:

$C = 2 \pi r$
$C = 2 \pi (4)$
$C = 8 \pi$

Using a calculator or estimating $\pi$ to 3.14, we get:

$C \approx 25.12$ inches.

However, we need to remember that we were asked for the circumference of the circle in the original units of inches, without rounding to the nearest hundredth. Therefore, we need to express the answer as a multiple of $\pi$:

$C = 8 \pi$ inches.

Using an approximation of $\pi \approx 3.14$, we can calculate:

$C \approx 25.12$ inches.

Therefore, the closest answer choice is C, 29.32 inches.

Find the Circumference of a Circle Inscribed in an Equilateral