Cone in a cylinder in a sphere

Geometry Level 3

Given the radius of a sphere $R$ and the height of a cyllinder $h$ , what is a formula for the volume $V$ of the cone in the picture?

V = \frac{1}{3}{\pi}h (R^{2} - h^{2})

V = \frac{1}{3}{\pi}R(h^{2} - \frac{R^{2}}{4}

V = \frac{1}{6}{\pi}h(R^{2} - h^{2})

V = \frac{1}{6}{\pi}h (R^{2} - \frac{h^{2}}{4})

V = \frac{4}{3}{\pi}R(h^{2} - \frac{R^{2}}{4}

2 solutions

Tomáš Hauser
Jun 19, 2018

$\begin{array}{l} r = \sqrt {{R^2}{\rm{ - }}{{\left( {\frac{h}{2}} \right)}^2}} = \sqrt {{R^2}{\rm{ - }}\frac{{{h^2}}}{4}} \\ V = \frac{1}{3}\pi {r^2}\left( {\frac{h}{2}} \right) = \frac{1}{6}\pi {r^2}h = \frac{1}{6}\pi {\left( {\sqrt {{R^2}{\rm{ - }}\frac{{{h^2}}}{4}} } \right)^2}h = \frac{1}{6}\pi h\left( {{R^2}{\rm{ - }}\frac{{{h^2}}}{4}} \right) \end{array}$

Edwin Gray
Jun 21, 2018

Let R = radius of the sphere.,r = radius of cylinder/cone, h = height of cylinder. Then R^2 = r^2 + (h/2)^2, V = (1/3) pi (r^2)(h/2). Substituting, and simplifying, V = (1/6) pi h[R^2 - h^2/4). Ed Gray

Cone in a cylinder in a sphere

2 solutions

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