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12345 [234]
3 years ago
15

CRITICAL THINKING Find the area between the three congruent tangent circles. The radius of each circle is 6 inches. Round your a

nswer to the nearest hundredth. Three congruent circles touch each other. The centers are marked as points A, B, and C. Equilateral triangle A B C is formed and the region inside triangle, but not in a circle, is shaded. The area is about a square inches.
Mathematics
1 answer:
Neporo4naja [7]3 years ago
8 0

Answer:

Area = 5.84in^2

Step-by-step explanation:

Given

Circles = 3

r = 6in --- The radius of each

Required

The area between them

See attachment for illustration of the question. (figure 1)

First, calculate the height of the equilateral triangle formed by the 3 radii (See figure 2)

Using Pythagoras theorem, we have:

12^2 = h^2 + 6^2

144 = h^2 + 36

Collect like terms

h^2 = 144 - 36

h^2 = 108

Take square roots

h = \sqrt{108

Expand

h = \sqrt{36 * 3

Split

h = \sqrt{36} * \sqrt{3

h = 6\sqrt{3

Now, the area of the equilateral triangle can be calculated using:

A = \frac{1}{2}bh

Where

h = 6\sqrt{3

b = 2r = 2 * 6 = 12

A = \frac{1}{2} * 12 * 6\sqrt 3

A = 6 * 6\sqrt 3

A_1 = 36\sqrt 3

Next, is to calculate the area of the sector formed by 2 radii in each circle (figure 3).

Since the radii formed an equilateral triangle, then the central angle will be 60. So:

A = \frac{\theta}{360} * \pi r^2

A = \frac{60}{360} * \pi 6^2

A = \frac{60}{360} * \pi * 36

A = \frac{60}{10} * \pi

A = 6\pi

For the three circles, the area is:

A_2 = 3 * 6\pi

A_2 = 18\pi

Subtract the areas of the sectors (A2) from the area of the equilateral triangle (A1), to get the area between them.

A = A_1 - A_2

A = 36\sqrt 3 - 18\pi

A = 36*1.7321 - 18 * 3.14

A = 5.8356

Approximate

A = 5.84in^2

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