The volume of a shape represents the three-dimensional space it occupies. You can also think of an object's volume as the amount of water (or air, sand, etc.) that would fit inside it to fill it completely. The most common units of volume are cubic centimeters (cm^{3}), cubic meters (m^{3}), cubic inches(in^{3}) and cubic feet (ft^{3}). This article will teach you how to calculate the volume of six different three-dimensional shapes commonly found on math tests, including cubes, spheres, and cones. You will find that many of these formulas are similar, which makes them even easier to remember. Try to memorize them throughout the article!

## Steps

### Method 1 of 6: Calculating the Volume of a Cube

#### Step 1. Recognize a cube

A cube is a three-dimensional shape that has six identical square faces. In other words, it's a box whose sides are all the same.

### A six-sided die is a good example of a cube, as are sugar cubes and children's letter blocks

#### Step 2. Learn the formula for finding the volume of a cube

Since all sides are equal, the formula for the volume of a cube is pretty easy: V = s^{3}, where V represents the volume and s is the length of one of the edges of the cube.

- To find s
^{3}, simply multiply the measure by itself three times: s^{3}= s * s * s

#### Step 3. Find the length of one side of the cube

Depending on your task, either the cube will come with the measure on one side written on it or you will have to measure it yourself. Keep in mind that because it's a cube, the measurements on all sides are the same, so it doesn't matter which one you measure.

### If you're not sure the shape is a cube, measure all sides to see if they're the same. If not, you will need to use the method to calculate the volume of a rectangular prism

**Step 4. Substitute the side measurement into the formula V = s ^{3} and calculate the volume**.

For example, if the measurement of the sides is 5 cm, you would write the formula as follows: V = (5 cm)^{3} = 5 cm * 5 cm * 5 cm = 125 cm^{3}. So, 125 cm^{3} is the volume of the cube!

#### Step 5. Record the answer in cubic units

In the example above, the length of the cube side was given in centimeters, so the volume should be given in cubic centimeters. If the side of the cube were 3 m, for example, the volume would be (3 m)^{3}, or V = 27 m^{3}.

### Method 2 of 6: Calculating the Volume of a Rectangular Prism

#### Step 1. Recognize a rectangular prism

A rectangular prism is a three-dimensional shape with six sides, all of which are rectangles. In other words, it's simply a three-dimensional rectangle or an ordinary box.

### A cube is just a rectangular prism whose sides of all rectangles are the same

#### Step 2. Learn the formula for finding the volume of a rectangular prism

The formula is V = c * l * a, where V = volume, c = length, l = width, and a = height.

#### Step 3. Find out the length value

The length is the longest side of the prism's bottom rectangular face. The value can be given in the figure or you will need to measure it to find it.

- Example: If the length of a rectangular prism is 4 cm, then c = 4 cm.
- Don't worry too much about knowing which side is the length, which is the width, etc. As long as you measure three different sides, the result will be the same regardless of the arrangement of the terms.

#### Step 4. Find the width value

The width of a rectangular prism is the shortest side of the prism's bottom rectangular face. Again, either the value will be given in the figure or you will have to measure it to find out.

- Example: if the width of a prism is 3 centimeters, then l = 4 cm.
- If you are measuring the rectangular prism with a ruler or measuring tape, remember to record all measurements in the same unit. Do not measure one side in centimeters and the other in inches; all measurements must be in the same unit!

#### Step 5. Find out the height value

Height is the distance from the surface or the lower rectangular face to the top of the prism. Locate this information in the figure or measure it yourself.

### Example: if the height of the rectangular prism is 6 cm, then a = 6 cm

#### Step 6. Substitute the dimensions of the rectangular prism into the formula and calculate the volume

Remember that V = c * l * a. Multiply the length, width and height. You can multiply them in any order, the result will be the same.

### In our example, c = 4, l = 3, and a = 6. Hence, V = 4 * 3 * 6, which equals 72

#### Step 7. Write down the answer in cubic units

As in our example the measurements were given in centimeters, the volume should be expressed as 72 cubic centimeters, or 72 cm^{3}.

- If the measurements were: length = 2 m, width = 4 m, and height = 8 m, the volume would be 2 m * 4 m * 8 m, which equals 64 m
^{3}.

### Method 3 of 6: Calculating the Volume of a Cylinder

#### Step 1. Learn to identify a cylinder

A cylinder is made up of two parallel circular bases and a closed, curved side surface that connects them.

### A can and a pile are good examples of cylinders

#### Step 2. Memorize the formula for calculating the volume of a cylinder

To calculate the volume of a cylinder, you need to know its height and the radius of its circular base (the distance from the center of the circle to its edge). The formula is V = πr^{2}h, where V represents the volume, r represents the radius of the circular base, h represents the height, and π is the value of the constant pi.

- In some geometry problems, the answer will have to be given in terms of π, but most of the time you will have to replace it with the value 3, 14. Ask your teacher which way he prefers.
- The formula for finding the volume of a cylinder is very similar to the formula for the volume of a rectangular prism: you will simply multiply the height of the shape by the surface area of its base. For the rectangular prism, this area was given by c * l, whereas for the cylinder, it is πr
^{2}, which represents area of a circle of radius r.

#### Step 3. Find the radius of the base

If the radius is given in the image, just use it. If the diameter is given instead of the radius, divide the value by 2 to obtain the radius measure (d = 2r).

#### Step 4. Measure the radius of the object if it is not given

Keep in mind that getting an accurate measurement of a circular solid can be a little tricky. One option is to measure the upper base of the cylinder with a ruler or tape. Measure the width of the cylinder at its widest part and divide the measurement found by 2 to get the radius.

- Another option is to measure the circumference of the cylinder using a tape measure. Once this is done, substitute the measure found in the formula: C (circumference) = 2πr. Divide the value of the circle by 2π (6, 28) and you will find the radius.
- For example, if you found a circumference of 8 centimeters, your radius would be 1.27 cm.
- If a really accurate measurement is needed, use both methods to ensure that the measurements are the same. If not, measure again. The circle method usually gives more accurate results.

#### Step 5. Calculate the area of the circular base

Substitute the radius of the base value into the formula A = πr^{2}. Just multiply the radius value by itself and then multiply the result by π. For example:

- If the radius of the circle is equal to 4 centimeters, the base area will be A = π4
^{2}. - 4
^{2}= 4 * 4 = 16. 16 * π (3, 14) = 50, 24 cm^{2} - If the base diameter is given instead of the radius, remember that d = 2r. Just divide the diameter by two to find the radius.

#### Step 6. Find the height value

The height of a cylinder is simply the distance between the two circular bases or the distance between the surface the object is on and its top. If the measurement is not given in the figure, measure it using a ruler or measuring tape.

#### Step 7. Multiply the base area by the height to find the volume

Or, you can directly substitute the values of the cylinder dimensions in the formula V = πr^{2}H. For our example, where the cylinder has a radius of 4 cm and a height of 10 cm, we have:

- V = π4
^{2}10 - π4
^{2}= 50, 24 - 50, 24 * 10 = 502, 4
- V = 502, 4

#### Step 8. Remember to present the answer in cubic units

In our example, measurements were given in centimeters, so the volume should be given in cubic centimeters: 502, 4 cm^{3}. If the cylinder were measured in inches, the volume would be expressed in cubic inches (in^{3}).

### Method 4 of 6: Calculating the Volume of a Regular Pyramid

#### Step 1. Understand what a regular pyramid is

A pyramid is a three-dimensional shape that has a polygon as its base and side faces that meet at a single point. A regular pyramid is one whose base polygon is regular, meaning that all sides and angles have the same measurement.

- Normally, we think of a pyramid as having a square base and triangular sides that meet at a common point, however the base of a pyramid can have 5, 6 or even 100 sides!
- A pyramid that has a circular base is called a cone, which will be covered in the next method.

#### Step 2. Learn the formula for calculating the volume of a regular pyramid

The formula is V = 1/3bh, where b is the area of the base of the pyramid and h is the height.

### The volume formula is the same for straight pyramids (those where the tip is above the center of the base) and oblique pyramids (those where the tip is not centered)

#### Step 3. Calculate the base area

The formula will depend on the number of sides the base of the pyramid has. Consider a pyramid with a square base whose sides are 6 centimeters long. Remember that the formula for the area of the square is A = s^{2}, where s is the measure of the sides. So we have that the base area is (6 cm)^{2} = 36 cm^{2}.

- The formula for the area of a triangle is: A = 1/2bh, where b is the base of the triangle and h is the height.
- You can find the area of any regular polygon using the formula A = 1/2pa, where A is the area, p is the perimeter of the shape, and a is the apothema - the distance from the center of the polygon to the midpoint of any of the its sides. This is a slightly more complex calculation that goes beyond the scope of this article. If you want to make the calculation easier, you can find great tips in this article.

#### Step 4. Find the height

In most cases the height will be indicated in the figure. Assume that the height of the pyramid is 10 cm.

#### Step 5. Multiply the base area by the height and divide the result by 3 to find the volume

Remember that the formula for volume is V =1/3bh. In our example, the base has an area of 36 and a height of 10, so the volume is: 36 * 10 * 1/3 = 120.

### If the pyramid had a pentagonal base with an area of 26 and a height of 8, the volume would be: 1/3 * 26 * 8 = 69, 33

#### Step 6. Don't forget to express the answer in cubic units

As the measurements in our example were given in centimeters, the volume should be expressed in cubic centimeters (120 cm^{3}). If measurements were given in meters, the volume should be expressed in cubic meters (m^{3}).

### Method 5 of 6: Calculating the Volume of a Cone

#### Step 1. Learn the properties of a cone

A cone is a three-dimensional solid with a circular base and a single vertex (the tip of the cone). Another way to look at it is as a pyramid with a circular base.

### If the cone's apex is directly above the center of the circular base, we say the cone is "straight". If the vertex is not directly above the center, it is called oblique

#### Step 2. Know the formula for finding the volume of a cone

The formula is V = 1/3πr^{2}h, where r represents the radius of the circular base, h represents height, and π is the constant pi, which can be rounded to 3, 14.

- The term πr
^{2}refers to the area of the circular base of the cone. Therefore, the formula for the volume of the cone is the same as the volume of the pyramid covered in the previous method!

#### Step 3. Calculate the area of the circular base

To do this, you need to know the radius of the base, which should be written in the figure. If the diameter is given, simply divide the value by 2, since the diameter is twice the radius (d = 2r). Then substitute the radius into the formula A = πr^{2} to calculate the area.

- Consider the radius to be 3 centimeters. Substituting this value in the formula we have: A = π3
^{2}. - 3
^{2}= 3 * 3 = 9. Hence, A = 9π. - H = 28.27 cm
^{2}.

#### Step 4. Find the height

The height of a cone is the vertical distance between the base and the vertex. Consider the height of the cone to be 5 centimeters.

#### Step 5. Multiply the base area by the height

In our example, the cone has a base area equal to 28.27 cm^{2} and height of 5 cm. Hence, bh = 28, 27 * 5 = 141, 35.

#### Step 6. Now multiply the result by 1/3 (or simply divide it by 3) to find the volume of the cone

In the previous step, we calculated the volume of the cylinder that would be formed if the cone walls extended to another circle. Dividing this value by 3 will give us the volume of the cone.

- In our example, 141, 35 * 1/3 = 47, 12.
- Doing otherwise, 1/3π3
^{2}5 = 47, 12.

#### Step 7. Present the answer in cubic units

Our cone was measured in centimeters, so its volume should be expressed in cubic centimeters: 47, 12 cm^{3}.

### Method 6 of 6: Calculating the Volume of a Sphere

#### Step 1. Recognize a sphere

The sphere is a perfectly round three-dimensional shape in which any point on its surface is the same distance from the center. In other words, a sphere is a ball-shaped object.

#### Step 2. Write down the formula for calculating the volume of a sphere

The formula is V = 4/3πr^{3} (read: four thirds of pi r cubed), where r is the radius of the sphere and π is the constant pi (3, 14).

#### Step 3. Find the radius of the sphere

If the radius is given in the figure, just use it. If given the diameter, simply divide the number by 2 to find the radius. As an example, consider the radius equal to 3 cm.

#### Step 4. Measure the radius if it is not given

If you need to measure a spherical object (such as a tennis ball) to find its radius, first find a tape long enough to loop around it. Then wrap the tape around the object at its widest part, marking the point where the tape overlaps itself. Divide this value by 2π or 6, 28 and you get the measure of the sphere's radius.

- For example, if you measure a ball and find that its circumference measures 18 centimeters, divide that number by 6.28 and you have the radius to measure 2.87 cm.
- Measuring a spherical object can be difficult, so try taking 3 measurements and using the average of the values found (suming them up and dividing them by 3) to ensure you use the most accurate result possible.
- For example, if the three measurements found are 18 cm, 17, 75 cm and 18, 2 cm, you would add these values (18 + 17, 5 + 18, 2 = 53, 95) and divide them by 3 (53, 95/3 = 17, 98). Use the average obtained in your calculations.

**Step 5. Cube the radius value to find r ^{3}**.

Just multiply it by itself three times, that is, r^{3} = r * r * r. In our example, the radius is 3 cm, so r^{3} = 3 * 3 * 3 = 27.

#### Step 6. Multiply the answer by 4/3

You can either use your calculator or do the math by hand. In our example, multiplying 27 by 4/3, we get 108/3, which is equal to 36.

#### Step 7. Multiply the answer by π to find the sphere's volume

Rounding the value of π to two decimal places is enough for most math problems (unless your teacher asks you to do it otherwise), so multiply the value found in the previous step by 3, 14 and you find the volume of the sphere.

### In our example, 36 * 3, 14 = 113, 09

#### Step 8. Present the answer in cubic units

Since the measurements in our example were given in centimeters, the answer should be V = 113.09 cubic centimeters (113.09 cm^{3}).