This lesson will help you understand clearly how to find the area between four touching circles. First, we need to show you what the area between four touching circles looks like.
The area in red above is the area between the four touching circles. To find it, label the figure as shown below. Notice that all the circles have the same radius that is x. Therefore, we are assuming here that the circles are congruent or equal.
Let x be the radius of each circle.
Let A be a sector or that
part of a circle bounded by two radii and their intercepted arc.
Therefore, B, C, and D are also sectors.
Here is the important or crucial observation you need to make to solve this problem.
area of sector A + area of sector B + area of sector C + area of sector D + area of red piece = area of the square
Since area of sector A = area of sector B = area of sector C = area of sector D
area of sector A + area of sector B + area of sector C + area of sector D = area sector A + area of sector A + area of sector A + area of sector A = 4 times area of sector A.
We can rewrite the part in blue and red as
4 times area of sector A + area of red piece = area of square
Area of red piece = area of square - 4 times area of sector A
Area of square = side times side = (x + x) times ( x + x) = 2x times 2x = 4x^{2}
4 times area of sector A = 4 times [ ( 3.14 times x^{2} ) / 4 ] = 3.14 times x^{2}
Therefore, area of red piece = 4x^{2} - 3.14x^{2}
= ( 4 - 3.14) x^{2}
= 0.86 x^{2}
Example #1
Suppose the radius is equal to 5 cm, what is the area between the four touching circles?
Area = 0.86x^{2}
Area = 0.86 times 5 cm times 5 cm
Area = 0.86 times 25 = 21.5 cm^{2}
Example #2
Suppose the radius is equal to 8 feet, what is the area between the four touching circles?
Area = 0.86x^{2}
Area = 0.86 times 8 feet times 8 feet
Area = 0.86 times 64 = 55.04 ft^{2}
Sep 30, 22 04:45 PM