Hello, quick question to double check my answers about calculating the area inside the overlappings of 2 different size circles.
The grey area is what I'm calculating: These are the formulas provided and a sheet of paper
Here is my working out, no angles were given, only 3 lengths. Points AC is a straight line.
To work out angle AEC, I used cosA = b^2+c^2-a^2 / 2bc
Then rearranged to get angle A which will be A = cos^-1 (b^2+c^2-a^2 / 2bc) -> cos^-1 (12^2+15^2-19^2 / 2x12x15) = 88.7 degrees (3 S.F)
Angle ECA will use the same formula. cos^-1 (b^2+c^2-a^2 / 2bc) -> cos^-1 (12^2+19^2-15^2 / 2x19x12) = 52.1 degrees (3 S.F)
Angle CAE will be 39.2 degrees since triangles add up to 180 degrees.
I then drew symmetrical lines to the bottom side, which will be equal since A and C are in the center of the circle.
I also put in a line directly down the center of the would-be shaded area.
I now know that angle ECF would be 2xECA since its symmetrical. so angle ECF is 52.1x2 = 104.2 degrees
CKF is 90 degrees so we can use SOHCAHTOA to find out the length of CK.
we have the hypotenuse, the angle KCF angle (52.1) and to find out the adjacent side which is CK
CAH = cos 52.1 = Adjacent / 12
rearrange to work out adjacent which will be cos 52.1 x 12 = Adjacent.
cos52.1 x 12 = 7.37, CK = 7.37m
We can now work out EF by finding the opposide side of angle KCF and doubling or use the Sine rule
By using the opposite side method, we can use trigonometry again.
sin 52.1 = opposite / 12 (hypotenuse)
sin 52.1 x 12 = opposite side (KF) = 9.47 * 2 = 18.94 m
to calculate the left-hand side of oval shaded area, I could calculate the area of the sector from Circle C (CEDF) then minus the area of the triangle EFC which will give me the segment of the circle.
To calculate the area of ECF triangle, we can use formula 1/2 x a x b x sin C or easier bxh / 2.
Area triangle = 18.94 x 7.37 / 2 = 69.8 m^2.
area of sector = pi x r^2 x angle/360
= pi x 12^2 x 104.2/360 = 130.9 m^2
area of segment = (area of sector) - (area of triangle)
area of segment = 130.9 - 69.8 = 61.1m^2
Now we have to calculate 2nd shaded area.
We need to figure out the length of AK since we already have the height.
AK can be used in trigonometry
angle KAF = 39.2
cos 39.2 = adajcent / 15
cos 39.2 x 15 =
= 11.6 m
area of the triangle (left side) is 11.6 x 18.94 / 2 = 109.1 m^2
sector = pi x 15^2 x (39.2x2)/360 = 153.9 m^2
segment = 153.9 - 109.1 = 44.8 m^2
the shaded area is 44.8 + 61.1 = 105.9 m ^2
If I did anything wrong, please correct me, thanks.
The grey area is what I'm calculating: These are the formulas provided and a sheet of paper
Here is my working out, no angles were given, only 3 lengths. Points AC is a straight line.
To work out angle AEC, I used cosA = b^2+c^2-a^2 / 2bc
Then rearranged to get angle A which will be A = cos^-1 (b^2+c^2-a^2 / 2bc) -> cos^-1 (12^2+15^2-19^2 / 2x12x15) = 88.7 degrees (3 S.F)
Angle ECA will use the same formula. cos^-1 (b^2+c^2-a^2 / 2bc) -> cos^-1 (12^2+19^2-15^2 / 2x19x12) = 52.1 degrees (3 S.F)
Angle CAE will be 39.2 degrees since triangles add up to 180 degrees.
I then drew symmetrical lines to the bottom side, which will be equal since A and C are in the center of the circle.
I also put in a line directly down the center of the would-be shaded area.
I now know that angle ECF would be 2xECA since its symmetrical. so angle ECF is 52.1x2 = 104.2 degrees
CKF is 90 degrees so we can use SOHCAHTOA to find out the length of CK.
we have the hypotenuse, the angle KCF angle (52.1) and to find out the adjacent side which is CK
CAH = cos 52.1 = Adjacent / 12
rearrange to work out adjacent which will be cos 52.1 x 12 = Adjacent.
cos52.1 x 12 = 7.37, CK = 7.37m
We can now work out EF by finding the opposide side of angle KCF and doubling or use the Sine rule
By using the opposite side method, we can use trigonometry again.
sin 52.1 = opposite / 12 (hypotenuse)
sin 52.1 x 12 = opposite side (KF) = 9.47 * 2 = 18.94 m
to calculate the left-hand side of oval shaded area, I could calculate the area of the sector from Circle C (CEDF) then minus the area of the triangle EFC which will give me the segment of the circle.
To calculate the area of ECF triangle, we can use formula 1/2 x a x b x sin C or easier bxh / 2.
Area triangle = 18.94 x 7.37 / 2 = 69.8 m^2.
area of sector = pi x r^2 x angle/360
= pi x 12^2 x 104.2/360 = 130.9 m^2
area of segment = (area of sector) - (area of triangle)
area of segment = 130.9 - 69.8 = 61.1m^2
Now we have to calculate 2nd shaded area.
We need to figure out the length of AK since we already have the height.
AK can be used in trigonometry
angle KAF = 39.2
cos 39.2 = adajcent / 15
cos 39.2 x 15 =
= 11.6 m
area of the triangle (left side) is 11.6 x 18.94 / 2 = 109.1 m^2
sector = pi x 15^2 x (39.2x2)/360 = 153.9 m^2
segment = 153.9 - 109.1 = 44.8 m^2
the shaded area is 44.8 + 61.1 = 105.9 m ^2
If I did anything wrong, please correct me, thanks.
