The common area to each circle is composed of the segments of each
circle formed by the line joining the intersection points of the circles.
Response:
- The area common to each circle is approximately 122.84 m²
How can the area of a segment be used to calculate the common area?
Given:
The diameter of the each circle, D = 20 m
The distance between the centers of the circle = The radius, R = [tex]\dfrac{20 \, m}{2}[/tex] = 10 m
By symmetry, and similar triangles, the line from the intersection of the
two circles bisect the line joining the centers.
Therefore, the angle formed by the two radii from a circle center to the
points of intersection of the circumference of the two circles are;
[tex]\theta = 2 \times arccos \left(\dfrac{5}{10} \right) = \mathbf{120^{\circ}}[/tex]
[tex]Area \ of \ a \ sector \ of \ a \ circle = \mathbf{\dfrac{\theta}{360^{\circ}} \times \pi \times R}[/tex]
[tex]Area \ of \ a \ segment \ of \ a \ circle \ is; \ \mathbf{R^2 \cdot \left(\dfrac{\theta \cdot \pi}{360 ^{\circ} } -\dfrac{sin(\theta)}{2} \right)}[/tex]
Where;
R = The radius of the circle
θ = The angle formed by the two radii
The area common to each circle is the sum of the two segments
created by the two circles by the line joining the point of intersection at
the circumference.
Therefore;
- [tex]The \ common \ area \ is; \ A = \mathbf{2 \times R^2 \cdot \left(\dfrac{\theta \cdot \pi}{360 ^{\circ} } -\dfrac{sin(\theta)}{2} \right)}[/tex]
Which gives;
[tex]A = 2 \times 10^2 \cdot \left(\dfrac{ 120 ^{\circ} \times \pi}{360 ^{\circ} } -\dfrac{sin(120 ^{\circ} )}{2} \right) = \mathbf{200 \times \left(\dfrac{\pi}{3} - \dfrac{\sqrt{3} }{4} \right)} \approx 122.84[/tex]
- The area common to each, A ≈ 122.84 m²
Learn more about segments and sectors of a circle here:
https://brainly.com/question/2501134
https://brainly.com/question/2284207
