A cap on a sphere includes a circular base. Without that base it's called a calotte


cap
calotte


In order to find the volume and surface area of a sphere's cap, here are the three variables we will need:

  • r is the radius of the sphere

  • a is the radius of the circular base of the cap

  • h is the distance from the top of the sphere
            to the centre of the circular base of the cap

    For simplification purposes we will first express a in terms of the other variables r and h.

    In the yellow right triangle we use the Pythagorean Theorem:

    r2 = a2 + (r - h)2
    a2 = r2 - (r - h)2
    a2 = r2 - (r2 - 2rh + h2)
    a2 = r2 - r2 + 2rh - h2
    a2 = 2rh - h2



    Surface Area of Cap:

    SA = area of outer side of curved part + area of circular base
    SA = 2πrh + πa2
    SA = 2πrh + π(2rh - h2)
    SA = 2πrh + 2πrh - πh2
    SA = 4πrh - πh2
    SA = πh(4r - h)



    Surface Area of Calotte:

    SA = area of outer side of curved part + area of circular base
    SA = 2πrh



    Volume of Cap:




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