How to find the volume of a polyhedron - Math

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Question

Find the volume of the following half cylinder.

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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.

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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.

Tap to reveal answer

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.

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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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Question

Find the volume of the following half cylinder.

Tap to reveal answer

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.

Tap to reveal answer

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.

Tap to reveal answer

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.

Tap to reveal answer

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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Question

Find the volume of the following half cylinder.

Tap to reveal answer

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.

Tap to reveal answer

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.

Tap to reveal answer

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.

Tap to reveal answer

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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Question

Find the volume of the following half cylinder.

Tap to reveal answer

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$$

← Didn't Know|Knew It →

Question

Find the volume of the following polyhedron.

Tap to reveal answer

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.

Tap to reveal answer

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 $$\sqrt{3}$m^3$ + 81 \pi $m^3$$

← Didn't Know|Knew It →

Question

Find the volume of the following polyhedron.

Tap to reveal answer

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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