A swimming pool is to be constructed in the space of partially overlapping identical circles. Each of the circles has a radius of 9 m, and each passes through the center of the other. Find the area of the swimming pool.
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http://www.analyzemath.com/Geometry/circles_problems.html
To find the area of the swimming pool, we first need to determine the shape formed by the overlapping circles. Since each circle passes through the center of the other, they form a shape known as a Vesica Piscis.
The Vesica Piscis is a lens-shaped figure with two circular arcs that overlap. The shape is symmetric and can be divided into two equal parts by its major axis, which is the line segment connecting the centers of the two circles.
Next, let's find the length of the major axis. The radius of each circle is 9 m, so the diameter, which is also the length of the chord of the circle formed by the major axis, is twice the radius, or 18 m.
The length of the major axis, which is also the chord of the overlapping circular arcs, can be found using the formula:
Length of major axis = 2 * sqrt(3) * radius
Substituting the radius value of 9 m:
Length of major axis = 2 * sqrt(3) * 9
= 18 * sqrt(3)
Now that we have the length of the major axis, we can find the area of the Vesica Piscis.
Area of Vesica Piscis = ((sqrt(3) * length of major axis)^2) / 4
Substituting the length of the major axis:
Area of Vesica Piscis = ((sqrt(3) * 18 * sqrt(3))^2) / 4
= (54^2 * 3) / 4
= (2916 * 3) / 4
= 8748 / 4
= 2187
Therefore, the area of the swimming pool, which is the area of the Vesica Piscis, is 2187 square meters.
To find the area of the swimming pool, we need to determine the overlapping area of the circles.
First, let's visualize the situation. We have two identical circles, both with a radius of 9 meters. Each of the circles passes through the center of the other circle, meaning they intersect.
To find the overlapping area, we can simply calculate the area of one of the circles and then subtract the area of the non-overlapping portion.
The formula to find the area of a circle is A = πr^2, where A is the area and r is the radius.
Using this formula, the area of one circle is:
A1 = π(9)^2 = 81π square meters
Since the circles have the same radius, the area of the other circle is also 81π square meters.
Now, let's determine the non-overlapping portion. Since the circles intersect, the overlapping area can be thought of as a lens shape. The lens shape can be divided into two equal segments by the line connecting the centers of the circles.
To find the area of the non-overlapping portion, we need to calculate the area of the entire lens shape and then divide it by 2.
The formula to find the area of a circular sector (the lens shape) is A = (θ/360) πr^2, where θ is the angle of the sector and r is the radius.
Since the circles have the same radius and they intersect at the center, the angle of each sector is 180 degrees (or π radians). Using this formula, the area of the lens shape is:
A2 = (π/360) (180) (9)^2 = 81π/2 square meters
However, this is the area of the entire lens shape. To find the non-overlapping portion, we need to divide it by 2:
A_non_overlapping = (81π/2) / 2 = 81π/4 square meters
Finally, we can find the overlapping area by subtracting the area of the non-overlapping portion from the area of a single circle:
A_overlapping = A1 - A_non_overlapping = 81π - (81π/4) = 81π(1 - 1/4) = 81π(3/4) = 243π/4 square meters
Hence, the area of the swimming pool, which is the overlapping area of the circles, is 243π/4 square meters.