The area of intersection of two circles can be computed algebraically or with calculus. Both methods require you to consider the shape of the overlapping region. Close inspection of this region reveals that it is a lens or planar structure with two curved surfaces. The area of the overlapping region can be calculated using the formula for lens area, which is applicable to symmetric or asymmetric lenses. Asymmetric lens regions occur when the circles have different sizes, while identical overlapping circles generate symmetric lenses. Careful substitution into the formula yields accurate results that match those derived through more complex methods.

Select variables to represent the properties of the overlapping circles. Let the radius of the first circle be R. Represent the radius of the second circle with r. The distance between the centres of the two circles is d.

Calculate T1, which is the first of three terms in the formula for the area of overlap between the circles. T1 is calculated using T1 = (r^2) x arccosine [({d^2} + {r^2} -- {R^2}) / {2 x d x r}]. For example, if the centers of two overlapping circles are separated by d = 10 cm, and both have the same radius r = R = 10 cm, then T1 = 104.7 = (10) x (10) x arccosine [({10 x 10} + {10 x 10} -- {10 x 10}) / {2 x 10 x 10}] = 100 x arccosine (0.5) = 100 x (60 degrees) = 100 x (pi/3 radians) where pi is approximately equal to 3.14.

Calculate T2, which is the second of the three terms required for determining the area of overlap between the circles. T2 is calculated using T2 = (R^2) x arccosine [({d^2} - {r^2} + {R^2}) / {2 x d x R}]. For example, if the centers of two overlapping circles are separated by d = 10 cm and both have the same radius r = R = 10 cm, then T2 = 104.7 = (10) x (10) x arccosine [({10 x 10} - {10 x 10} + {10 x 10}) / {2 x 10 x 10}] = 100 x arccosine (0.5) = 100 x (60 degrees) = 100 x (pi/3 radians) where pi is approximately equal to 3.14.

Calculate T3, the third term in the formula for the area of overlap between the circles. T3 is calculated using T3 = (0.5) x (square root [(r + R - d) x (r -- R + d) x (R -- r + d) x (R + r + d)]). For example, if the centers of two overlapping circles are separated by d = 10 cm, and both have the same radius r = R = 10 cm, then T3 = 86.6 = (0.5) x (square root [(10 + 10 - 10) x (10 -- 10 + 10) x (10 -- 10 + 10) x (10 + 10 + 10)]) = (0.5) x (square root [30000]) = 0.5 x 173.2.

Find the area (A) of overlap between the intersecting circles using the formula A = T1 + T2 -- T3. For example, if the centres of two overlapping circles are separated by d = 10cm and both have the same radius r = R = 10cm, then the area of overlap = 122.8cm^2 = T1 + T2 - T3 = 104.7 + 104.7 -- 86.6.

#### Warning

Most scientific calculators default to the degrees (DEG) mode. If the lens formula is applied while the calculator is operating in the degrees mode, the arccosine term (found in steps 2 and 3) will represent a number of degrees. These degrees must be converted to radians to avoid an error: 180 degrees = pi radians = approximately 3.14 radians. It is recommended that the calculations be done with the calculator operating in the radians (RAD) mode, so the conversion of degrees to radians is performed automatically.

#### Tips and warnings

- Most scientific calculators default to the degrees (DEG) mode. If the lens formula is applied while the calculator is operating in the degrees mode, the arccosine term (found in steps 2 and 3) will represent a number of degrees. These degrees must be converted to radians to avoid an error: 180 degrees = pi radians = approximately 3.14 radians. It is recommended that the calculations be done with the calculator operating in the radians (RAD) mode, so the conversion of degrees to radians is performed automatically.