Geometry - How to find the volume of a cube

A sphere with a radius of is cut out of a cube that has a side edge of . What is the volume of the resulting shape?

Explanation

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(2)^3=\frac{32}{3}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=4^3=64$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=64-\frac{32}{3}\pi=30.49$

Make sure to round to places after the decimal.

The side length of a cube is ft.

What is the volume?

Explanation

The volume of a cube is

So with a side length of 2 ft, the volume is

$V = 2^{3}, ft^3 = 8, ft^3$

A sphere with a radius of is cut out of a cube that has a side length of . What is the volume of the resulting figure?

Explanation

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(1)^3=\frac{4}{3}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=5^3=125$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=125-\frac{4}{3}\pi=120.81$

Make sure to round to places after the decimal.

A rectangular prism with a square base is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Explanation

Start by finding the volume of the cube.

$\text{Volume of Cube}=\text{edge}^3$

$\text{Volume of Cube}=13^3=2197$

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=2 \times 2 \times 13=52$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=2197- 52=2145$

A rectangular prism with a square base is cut out of a cube as shown by the figure below.

Find the volume of the fgiure.

Explanation

Start by finding the volume of the cube.

$\text{Volume of Cube}=\text{edge}^3$

$\text{Volume of Cube}=10^3=1000$

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=8 \times 8 \times 10=640$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=1000- 640=360$

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

Explanation

If we let represent the length of an edge on the smaller cube, its volume is $x^{3}$ .

The larger cube has edges three times as long, so the length can be represented as . The volume is $(3x)^{3}$ , which is $27x^{3}$ .

The large cube's volume of $27x^{3}$ is 27 times as large as the small cube's volume of $x^{3}$ .

A sphere with a radius of $\frac{2}{3}$ is cut out of a cube that has a side length of . What is the volume of the resulting figure?

Explanation

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{2}{3})^3=\frac{32}{81}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=3^3=27$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=27-\frac{32}{81}\pi=25.76$

Make sure to round to places after the decimal.

A rectangular prism with a square base is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Explanation

Start by finding the volume of the cube.

$\text{Volume of Cube}=\text{edge}^3$

$\text{Volume of Cube}=9^3=729$

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=3\times3 \times 9=81$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=729- 81=648$

A rectangular prism with a square base is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Explanation

Start by finding the volume of the cube.

$\text{Volume of Cube}=\text{edge}^3$

$\text{Volume of Cube}=15^3=3375$

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=10 \times 10 \times 15=1500$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=3375- 1500=1875$

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Explanation

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

Plug in the given side length of the cube to find the volume.

$\text{Volume of Cube}=5^3=125$

Next, recall how to find the volume of a cylinder.

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{3}{2}=1.5$

Find the volume of the cylinder.

$\text{Volume of Cylinder}=\pi\times 1.5^2 \times 5=11.25\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=125-11.25\pi=89.66$

Make sure to round to places after the decimal.

How to find the volume of a cube

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