Thr figure below is made up of four congruent circles Each circle is tsngent to two of the other 3 circles as shown. If the radius of each circle is 6 units, what is the area of the shaded region inthe figure below? Express your answer as adecimal to the nearest hundredth.
I can't see a figure, but I assume the 4 circles are arranged in a square. Each circle has a diameter of 12 inches, so let's consider the 12" square forming one quarter of the figure.
Area of square is 144; area of circle in square is 36pi. Each corner outside the square thus has area 1/4 (144-36pi) = 36-9pi.
The center area, consisting of 4 of these areas, thus has area 144-36pi.
SUrely the shaded area can be figured from these data.
To find the area of the shaded region, let's break it down step by step.
Step 1: Find the area of one circle.
The formula to find the area of a circle is A = πr², where A represents the area and r represents the radius. In this case, each circle has a radius of 6 units, so the area of one circle can be calculated as:
A = π(6)² = 36π square units.
Step 2: Find the area of the unshaded region.
The unshaded region consists of the four circles arranged in a pattern. It forms a square with side length equal to the sum of the diameters of the circles. The diameter of a circle is twice the radius, so the side length of the square is 2(6) + 2(6) = 24 units. The area of a square is calculated by multiplying the side length by itself, so the area of the unshaded region is:
A_unshaded = (24)² = 576 square units.
Step 3: Find the area of the shaded region.
The shaded region consists of the area outside the four congruent circles but inside the square. To find this area, subtract the area of the four circles from the area of the unshaded region:
A_shaded = A_unshaded - 4(A_circle).
Note that we have four circles, so we subtract four times the area of one circle. Plugging in the values, we get:
A_shaded = 576 - 4(36π).
Step 4: Simplify and approximate the answer.
Calculating the expression, we have:
A_shaded = 576 - 144π.
Using a calculator, if we approximate π to the nearest hundredth, which is 3.14, we can find the numerical value of A_shaded. Let's substitute π = 3.14 and calculate the final answer:
A_shaded = 576 - 144(3.14) ≈ 576 - 452.16 ≈ 123.84 square units.
Therefore, the approximate area of the shaded region is 123.84 square units.