How to find the volume of a polyhedron - High School Math

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Question

Find the volume of the following half cylinder.

Answer

The formula for the volume of a half-cylinder is:

$V = \frac{1}{2} (\pi r^2 h)$

where is the radius of the base and is the length of the height.

Plugging in our values, we get:

$V = \frac{1}{2} \pi (5m)^2 (15m)$

$V = 187.5 \pi m^3$

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Question

Find the volume of the following polyhedron.

Answer

The formula for the volume of the polyhedron is:

$V = V_{cone}+\frac{1}{2}V_{sphere}$

$V = \frac{\pi r^2 h}{3} +\frac{1}{2} \frac{4\pi r^3}{3}$

$V = \frac{\pi r^2 h}{3} +\frac{2\pi r^3}{3}$

Where is the radius of the cone, is the height of the cone, and is the radius of the sphere.

Use the formula for a triangle to find the length of the radius:

$a-a\sqrt{2}-2a$

$3\sqrt{3}m-9m-6\sqrt{3}m$

Plugging in our values, we get:

$V = \frac{\pi (3\sqrt{3}m)^29m}{3}+\frac{2 \pi (3\sqrt{3}m)^3}{3}$

$V = \frac{\pi (27m^2)9m}{3}+\frac{2 \pi (81\sqrt{3}m^3)}{3}$

$V = 81 \pi m^3+54\sqrt{3} \pi m^3$

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Question

Find the volume of the following polyhedron.

Answer

The formula for the volume of the polyhedron is:

$V = \frac{\pi r^2 h}{3}+\pi r^2 h$

where is the radius of the cone, is the height of the cone, is the radius of the cylinder, and is the height of the cylinder.

Use the formula for a triangle to find the length of the radius and height of the cone:

$a-a\sqrt{3}-2a$

$3m-3\sqrt{3}m-6m$

Plugging in our values, we get:

$V = \frac{\pi (3m)^2 (3\sqrt{3}m)}{3}+\pi (3m)^2 (9m)$

$V = 9 \pi \sqrt{3}m^3 + 81 \pi m^3$

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Question

Find the volume of the following polyhedron.

Answer

The formula for the volume of the polyhedron is:

$V = (cone) + \frac{1}{2}(sphere)$

$V = \frac{1}{3}(\pi r^2)(h)+\frac{1}{2}\left(\frac{4\pi r^3}{3}\right)$

where is the radius of the polyhedron and is the height of the cone.

Use the formula for a triangle to find the length of the radius and height:

$a-a-a\sqrt{2}$

$4\sqrt{2}m-4\sqrt{2}m-8m$

Plugging in our values, we get:

$V = \frac{1}{3}(\pi (4\sqrt{2}m)^2)(4\sqrt{2}m)+\frac{1}{2}\left(\frac{4\pi (4\sqrt{2}m)^3}{3}\right)$

$V = 128 \sqrt{2} \pi m^3$

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