The volume of an object represents the three-dimensional space occupied by the object. You can also think of volume as how much water (or air, or sand, etc.) a shape can hold if the shape is completely filled. The unit commonly used for volume is the cubic centimeter (cm3), cubic meters (m3), cubic inches (in3), and cubic feet(ft3). This article will teach you how to calculate the volumes of six different three-dimensional shapes that are often found on math exams, including cubes, spheres, and cones. You may notice that many of these volume formulas share something in common so they are easy to remember. See if you can figure this out!
Info at a glance: Calculating the Volume of Common Forms
- For a solid cube or square, measure the length, width, and height and then multiply them all together to get the volume. See pictures and details.
- Measure the height of the tube and its base radius. Use this radius to find the base area using the formula r2, then multiply the result by the height of the tube. See pictures and details.
- A standard pyramid has a volume equal to x base area x height. See pictures and details.
- The volume of a cone can be calculated using the formula r2h, where r is the radius of the base and h is the height of the cone. See pictures and details.
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To measure the volume of a sphere, all you need is its radius r. Plug this value into the formula 4/3r3. See pictures and details.
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Method 1 of 6: Calculating the Volume of a Cube
Calculate Volume Step 1 Step 1. Know the shape of a cube
A cube is a three-dimensional shape that has six equal-sized square sides. In other words, a cube is a box with all sides the same size.
A 6-sided die is an example of a cube you might find in your home. Sugar blocks, and children's toy letter blocks are usually cubes too
Calculate Volume Step 2 Step 2. Learn the formula for the volume of a cube
The formula is simple V= s3, where V represents the volume and s represents the side length of the cube.
To find s3, multiply a by its own value 3 times: s3 = s * s * s
Calculate Volume Step 3 Step 3. Measure the length of one side of the cube
Depending on your assignment, the cube may already have this information captioned, or you'll need to measure the length of the sides with a ruler. Keep in mind that since this is a cube, all of the side lengths will be the same so it doesn't matter which side you measure.
If you're not 100% sure that the shape you have is a cube, measure each side to see if it's the same size. If they are not the same, you must use the method below to Calculate Block Volume
Calculate Volume Step 4 Step 4. Plug the side lengths into the formula V = s3 and count.
For example, if the length of the sides of your cube is 5 inches, then you would write the formula like this: V = (5 in)3. 5 in * 5 in * 5 in = 125 in3, that's the volume of our cube!
Calculate Volume Step 5 Step 5. Express the result in cubic units
In the example above, the side lengths of our cube are measured in inches, so the unit of volume is in cubic inches. If the length of the side is 3 centimeters, for example, the volume is V = (3 cm)3, or V = 27 cm3.
Method 2 of 6: Calculating Block Volume
Calculate Volume Step 6 Step 1. Know the shape of a block
A block, also called a rectangular prism, is a three-dimensional shape with six sides that are all rectangular. In other words, the block is a three-dimensional rectangular shape, or the shape of a box.
A cube is just a special block with all sides the same size
Calculate Volume Step 7 Step 2. Learn the formula for calculating the volume of a cuboid
The formula for the volume of a cuboid is Volume = length * width * height, or V = plt.
Calculate Volume Step 8 Step 3. Find the length of the block
This length is the longest part of the side of a beam that is parallel to the surface on which the beam is placed. This length may already be given in the diagram, or you may have to measure it with a ruler or tape measure.
- Example: The length of this block is 4 inches, so p = 4 in.
- Don't worry too much about which side is the length, width, and height. As long as you use three different measurements, the end result will be the same, regardless of how you order them.
Calculate Volume Step 9 Step 4. Find the width of the beam
The beam width is the measurement of the shorter side of the solid parallel to where the beam is placed. Again, look for a label on the chart that indicates the width, or measure it yourself with a ruler or tape measure.
- Example: The width of this block is 3 inches, so l = 3 in.
- If you are measuring blocks with a ruler or tape measure, make sure you do so using the same units. Don't measure one side in inches and the other in centimeters; all measurements must use the same units!
Calculate Volume Step 10 Step 5. Find the height of the block
This height is the distance from the surface of the beam placed to the top of the beam. Look up the height information in your chart, or measure yourself with a ruler or tape measure.
Example: The height of this block is 6 inches, so t = 6 in
Calculate Volume Step 11 Step 6. Plug the cuboid measurements into the volume formula and calculate them
Remember that V = plt.
In our example, p = 4, l = 3, and t = 6. Therefore, V = 4 * 3 * 6, or 72
Calculate Volume Step 12 Step 7. Make sure you write down the result in cubic units
Since our sample block is measured in inches, its volume must be written as 72 cubic inches, or 72 in3.
If our cuboid's measurements are: length = 2 cm, width = 4 cm, and height = 8 cm, then the volume of the block is 2 cm * 4 cm * 8 cm, or 64 cm3.
Method 3 of 6: Calculating Tube Volume
Calculate Volume Step 13 Step 1. Identify the shape of a tube
A tube is a three-dimensional shape with two identical flat ends that are circular in shape, and a curved side joining the two.
A can is an example of a tube, as are AA or AAA batteries
Calculate Volume Step 14 Step 2. Remember the formula for the volume of a cylinder
To calculate the volume of a cylinder, you need to know the height and radius of the base circle (the distance from the center of the circle to the edge) at the top and bottom. The formula is V = r2t, where V is the volume, r is the radius of the base circle, t is the height, and is the constant value of pi.
- In some geometry problems, the answer will be about pi, but in most cases, we can round pi to 3, 14. Confirm this with your instructor to see which one he prefers.
- The formula for finding the volume of a cylinder is actually very similar to the formula for the volume of a cuboid: you just multiply the height of the shape by the surface area of the base. In the cuboid formula, this surface area is p * l, while for a cylinder it is r2, i.e. the area of a circle with radius r.
Calculate Volume Step 15 Step 3. Find the base radius
If given in the diagram, use the value. If the diameter is given instead of the radius, all you have to do is divide by 2 to find out the value of the radius (d = 2r).
Calculate Volume Step 16 Step 4. Measure the object if a radius is not given
Be aware that measuring the tube precisely can be quite difficult. One way is to measure the bottom of the tube pointing up with a ruler or measuring tape. Do your best to measure the width of the cylinder at its widest, and divide by 2 to find the radius.
- Another option for measuring the circumference of a tube (the distance around it) is to use a tape measure or a piece of string that you can mark and measure the length with a ruler. Then, plug that measurement into the formula C (circumference) = 2πr. Divide the circumference by 2π (6.28) and you'll get the radius.
- For example, if the circumference you are measuring is 8 inches, then the radius is 1.27 inches.
- If you really need accurate measurements, you can use both methods to ensure that your measurements are the same. If not, double check both. The circumference method usually gives more accurate results.
Calculate Volume Step 17 Step 5. Calculate the area of the base circle
Plug the base radius value into the r. formula2. Then, multiply the radius by itself once, and multiply the result again by. As an example:
- If the radius of your circle is 4 inches, then the base area is A = 42.
- 42 = 4 * 4, or 16. 16 * (3.14) = 50.24 inches2
- If the diameter of the base is given instead of the radius, remember that d = 2r. You just have to divide the diameter in half to find the radius.
Calculate Volume Step 18 Step 6. Find the height of the tube
This is the distance between the two halves of the circle, or the distance from the surface on which the tube is placed. Look for a label on your diagram indicating the height of the tube, or measure it with a ruler or tape measure.
Calculate Volume Step 19 Step 7. Multiply the area of the base by the height of the cylinder to find the volume
Or you can skip one step and enter the tube dimension values into the formula V = r2t. For our example with a tube that has a radius of 4 inches and a height of 10 inches:
- V = 4210
- 42 = 50, 24
- 50.24 * 10 = 502, 4
- V = 502, 4
Calculate Volume Step 20 Step 8. Remember to state your answer in cubic units
Our sample tube is measured in inches, so its volume must be expressed in cubic inches: V = 502.4 in3. If our cylinder is measured in centimeters, then its volume will be expressed in cubic centimeters (cm3).
Method 4 of 6: Calculating the Volume of an Ordinary Pyramid
Calculate Volume Step 21 Step 1. Understand what a regular pyramid is
A pyramid is a three-dimensional shape with a polygon as its base, and lateral sides that join in an axis (the vertex of the pyramid). A regular pyramid is a pyramid where the base is a standard polygon, meaning that all the sides of the polygon are equal in length, and all the angles are the same.
- We usually think of a pyramid as having a square base, with sides that culminate to a point, but actually the base of a pyramid can have 5, 6, or even 100 sides!
- A pyramid with a circular base is called a cone, which will be discussed in the next method.
Calculate Volume Step 22 Step 2. Learn the formula for calculating the volume of an ordinary pyramid
This formula is V = 1/3bt, where b is the area of the base of the pyramid (the shape of the polygon below it) and t is the height of the pyramid, or the vertical distance from the base to the apex.
The formula for the volume of a right pyramid is the same, where the vertex is directly above the center of the base, and for an oblique pyramid, where the vertex is not in the middle
Calculate Volume Step 23 Step 3. Calculate the base area
The formula for this will depend on the number of sides that the base of a pyramid has. In the pyramid in our diagram, the base is a square with sides 6 inches long. Remember that the formula for the area of a square is A = s2, where s is the side length. So, for this pyramid, the base area is (6 in) 2, or 36 in2.
- The formula for the area of a triangle is: A = 1/2bt, where b is the base of the triangle and t is the height.
- You can find the area of a standard polygon using the formula A = 1/2pa, where A is the area, p is the perimeter of the shape, and a is the apothem, or the distance from the shape's midpoint to the midpoint of one of its sides. This is a more complex calculation that won't be covered in this article, but you can visit the article Calculating the Area of a Polygon to learn some good instructions on how to use it. Or, you can simplify this process and look for a Polygon Calculator online.
Calculate Volume Step 24 Step 4. Find the height of the pyramid
In most cases, this will be shown in the diagram. In our example, the height of the pyramid is 10 inches.
Calculate Volume Step 25 Step 5. Multiply the area of the base of the pyramid by its height, and divide by 3 to find the volume
Remember that the volume formula is V = 1/3bt. In our example pyramid, which has an area of 36 and a height of 10, the volume is: 36 * 10 * 1/3, or 120.
If we use a different pyramid, for example one that has a pentago-shaped base with an area of 26 and a height of 8, the volume will be: 1/3 * 26 * 8 = 69, 33
Calculate Volume Step 26 Step 6. Remember to state your answer in cubic units
The measurements in our example pyramid are in inches, so the volume must be expressed in cubic inches, 120. If our pyramid is measured in meters, the volume must be expressed in cubic meters (m3).
Method 5 of 6: Calculating the Volume of a Cone
Calculate Volume Step 27 Step 1. Learn the shape of the cone
A cone is a 3-dimensional shape with a circular base and a vertex. Another way to think of it is to think of the cone as a pyramid with a circular base.
If the vertex of the cone is exactly in the center of the circle, then the cone is a "true cone". If the vertex is not exactly in the middle, then the cone is called an "oblique cone." Fortunately, the formula for calculating the volume of both is the same
Calculate Volume Step 28 Step 2. Master the formula for calculating the volume of a cone
The formula is V = 1/3πr2t, where r is the radius of the circular base of the cone, where t is the height, and is the constant pi, which is rounded up to 3.14.
r. part2 from the formula refers to the area of the base of the circular cone. Therefore, the formula for the volume of a cone is 1/3bt, just like the formula for the volume of a pyramid in the previous method!
Calculate Volume Step 29 Step 3. Calculate the area of the circular base of the cone
To do this, you need to know the radius, which should already be written in your diagram. If you are given only the diameter, divide that value by 2, because the diameter is 2 times the radius (d = 2r). Then enter the radius value into the formula A = r2 to calculate the area.
- In the example in the diagram, the radius of the base of the cone is 3 inches. When we plug it into the formula, then: A = 32.
- 32 = 3 *3, or 0, so A = 9π.
- A = 28, 27 in2
Calculate Volume Step 30 Step 4. Find the height of the cone
This is the vertical distance between the base of the cone and its apex. In our example, the height of the cone is 5 inches.
Calculate Volume Step 31 Step 5. Multiply the height of the cone by the area of the base
In our example, this area is 28.27 inches2 and the height is 5 inches, so bt = 28, 27 * 5 = 141, 35.
Calculate Volume Step 32 Step 6. Now multiply the result by 1/3 (or you can divide by 3) to find the volume of the cone
In the step above, we calculated the volume of the cylinder that would form if the walls of the cone extended straight into another circle instead of narrowing to a point. Dividing by 3 gives the volume of the cone itself.
- In our example, 141, 35 * 1/3 = 47, 12, this is the volume of the cone.
- Alternatively, 1/3π325 = 47, 12
Calculate Volume Step 33 Step 7. Remember to state your answer in cubic units
Our cone is measured in inches, so its volume must be expressed in cubic inches: 47.12 inches3.
Method 6 of 6: Calculating the Volume of a Ball
Calculate Volume Step 34 Step 1. Find out the shape
A sphere is a perfectly spherical three-dimensional object, where every point on its surface is the same distance from its center. In other words, what is included here is spherical objects.
Calculate Volume Step 35 Step 2. Learn the formula for the volume of a sphere
The formula for the volume of this sphere is V = 4/3πr3 (read: "four-thirds pi r-cube") where r is the radius of the sphere, and is the pin constant (3, 14).
Calculate Volume Step 36 Step 3. Find the radius of the sphere
If the radius is given, then finding r is just an easy matter. If the diameter is given, you must divide by 2 to find the radius value. For example, the radius of the sphere in our diagram is 3 inches.
Calculate Volume Step 37 Step 4. Measure the ball if the radius is unknown
If you need to measure a spherical object (such as a tennis ball) to find its radius, first take a string large enough to wrap around the object. Then, loop around the object at its widest point and mark where the string touches the end again. Then, measure the string with a ruler to find its outer circumference. Divide this value by 2π, or 6, 28, and you get the radius of the sphere.
- For example, if you measure a sphere and find that the outer circumference is 18 inches, divide it by 6.28 and you get a radius of 2.87 inches.
- Measuring spherical objects can be a little tricky, so make sure you measure 3 different times, and take the average (add all three measurements, then divide by 3) to make sure you get the most accurate value.
- For example, if your outer circumference measurements are 18 inches, 17.75 inches, and 18.2 inches, add them all up (18 + 17.5 + 18, 2 = 53.95) and divide the result by 3 (53.95/3 = 17, 98). Use this average in your volume calculations.
Calculate Volume Step 38 Step 5. Cube the radius to find r3.
This means you have to multiply it by the number itself 3 times, so r3 = r * r * r. In our example, r = 3, so r3 = 3 * 3 * 3, or 27.
Calculate Volume Step 39 Step 6. Now multiply your answer by 4/3
You can use a calculator, or you can calculate it manually and simplify the fraction. In our example, multiplying 27 by 4/3 = 108/3, or 36.
Calculate Volume Step 40 Step 7. Multiply the result by to find the volume of the sphere
The final step in calculating the volume is to multiply the result by. Rounding to two digits is usually sufficient for most math problems (unless your teacher says otherwise), so multiply by 3, 14 and you'll find the answer.
In our example, 36 * 3, 14 = 113, 09
Calculate Volume Step 41 Step 8. Express your answer in cubic units
In our example, the radius of the sphere is measured in inches, so our real answer is V = 113.09 cubic inches (113.09 inches).3).
