A spherical cap is a part of a sphere that is obtained by cutting it with a plane. It is the section of a sphere that extends above the sphere's plane and formed when a plane cuts off a part of a sphere. The base area, height, and sphere radius are all the values that are required to calculate the volume of a spherical cap.
Spherical Cap Volume Formula
Below is the formula for Spherical Cap Volume.
⇒ V = (1/3)π(3R - h)h2
where,
R is the radius of the sphere,
h is the height of the spherical cap,
π is a constant with a value of 22/7.
Using Pythagoras theorem, we can say that (R - h)2 + a2 = R2. So, the formula can also be written as,
V = (1/6)πh(3a2 + h2)
Here, a denotes the radius of spherical cap.
Sample Problems on Spherical Cap Volume
Problem 1. Find the volume of the spherical cap if the radius of the sphere is 7 m and the height of the cap is 10 m.
Solution:
We have, r = 7 and h = 10.
Using the formula we have,
V = (1/3)π(3R - h)h2
= (1/3) (22/7) (3(7) - 10) (102)
= (1/3) (22/7) (11) (100)
= 1152 cu. m
Problem 2. Find the volume of the spherical cap if the radius of the sphere is 5 m and the height of the cap is 5 m.
Solution:
We have, r = 5 and h = 5.
Using the formula we have,
V = (1/3)π(3R - h)h2
= (1/3) (22/7) (3(5) - 5) (52)
= (1/3) (22/7) (10) (25)
= 261.8 cu. m
Problem 3. Find the volume of the spherical cap if the radius of the sphere is 7.5 m and the height of the cap is 12 m.
Solution:
We have, r = 7.5 and h = 12.
Using the formula we have,
V = (1/3)π(3R - h)h2
= (1/3) (22/7) (3(7.5) - 5) (122)
= (1/3) (22/7) (17.5) (144)
= 1583.4 cu. m
Problem 4. Find the radius of the sphere if the height and volume of the cap are 15 m and 2120.6 cu. m respectively.
Solution:
We have, V = 2120.6 and h = 15.
Using the formula we have,
V = (1/3)π(3R - h)h2
=> 2120.6 = (1/3) (22/7) (3R - 15) (152)
=> 2120.6 = (1/3) (22/7) (3R - 15) (225)
=> 3R - 15 = 9
=> 3R = 24
=> R = 8 m
Problem 5. Find the radius of the spherical cap if the height and volume of the cap are 6.5 m and 1305.2 cu. m respectively.
Solution:
We have, V = 1305.2 and h = 6.5.
Using the formula we have,
V = (1/3)π(3R - h)h2
=> 1305.2 = (1/3) (22/7) (3R - 15) (6.5)2
=> 1305.2 = (1/3) (22/7) (3R - 6.5) (42.25)
=> 3R - 6.5 = 29.5
=> 3R = 36
=> R = 12 m
Now using the formula (R - h)2 + a2 = R2, we have
a2 = R2 - (R - h)2
a2 = 122 - (12 - 6.5)2
a2 = 144 - 30.25
a2 = 113.75
a = 10.67 m
Problem 6. Find the volume of a spherical cap if its radius is 7 m and height is 14 m.
Solution:
We have, a = 7 and h = 14.
Using the formula we have,
V = (1/6)πh(3a2 + h2)
= (1/6) (22/7) (14) (3 (7)2 + 142)
= (1/6) (22/7) (14) (343)
= 2514.3 cu. m
Problem 7. Find the volume of a spherical cap if its radius is 4.21 m and height is 9.54 m.
Solution:
We have, a = 4.21 and h = 9.54.
Using the formula we have,
V = (1/6)πh(3a2 + h2)
= (1/6) (22/7) (9.54) (3 (4.21)2 + 9.542)
= (1/6) (22/7) (14) (144.183)
= 720.2 cu. m
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Practice Problems - Spherical Cap Volume Formula
1: Calculate the volume of a spherical cap with a height of 3 units and the radius of the sphere is 12 units.
2: If the volume of a spherical cap is 512π/3 cubic units and the height of the cap is 8 units, find the radius of the sphere.
3: Calculate the volume of a spherical cap with a height of 1 units and the radius of the sphere is 13 units.
4: Calculate the volume of a spherical cap with a height of 6 units and the radius of the sphere is 21 units.
5: Calculate the volume of a spherical cap with a height of 3.5 units and the radius of the sphere is 1.2 units.
6: Calculate the volume of a spherical cap with a height of 2.5 units and the radius of the sphere is 16 units.
7: Calculate the volume of a spherical cap with a height of 1.5 units and the radius of the sphere is 15 units.
8: Calculate the volume of a spherical cap with a height of 0.5 units and the radius of the sphere is 14 units.
9: Calculate the volume of a spherical cap with a height of 3.4 units and the radius of the sphere is 5 units.
10: Calculate the volume of a spherical cap with a height of 2.5 units and the radius of the sphere is 1 units.
Summary
The volume of a spherical cap is given by the formula