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The volume of a shape indicates the amount of space it occupies in three-dimensional space. You can visualize the volume of a shape as the amount of water (or air, sand, etc.) that the shape could contain when filled with such substances. Common units for volume include cubic centimeters (cm3), cubic meters (m3), cubic inches (in3), and cubic feet (ft3). This article will guide you through calculating the volume of six common 3D shapes found in math exams, including cubes, rectangular prisms, cylinders, pyramids, cones, and spheres. You’ll notice that the volume formulas share some similarities, which can help you remember them. Follow the steps below to see if you can identify those common points!
Steps
Calculating the Volume of a Cube
Recognizing a Cube. A cube is a three-dimensional shape with six square faces. In other words, it’s a box with all sides of equal length.
- A six-sided die is an example of a cube you can find at home. A compressed sugar cube or children's alphabet blocks are also often cube-shaped.
Formula for Calculating the Volume of a Cube. Since all edges of a cube are equal, the formula for calculating its volume is straightforward. It is: V = s3, where V is the volume and s is the edge length of the cube.
- To find s3, you simply multiply s by itself three times, i.e., s3 = s * s * s
Finding the Length of One Edge of a Cube. Depending on the case, the problem might provide this value, or you may need to measure the cube’s edge using a ruler. Since this is a cube, all edges are equal, so you only need to measure one edge.
- If you're not 100% sure that the shape you're measuring is a cube, measure all the edges and see if the values match. If they don't, you’ll need to apply the formula for calculating the volume of a rectangular prism, which will be explained in the next section.
Substitute the Measured Length into the Formula V = s3 and Calculate. For example, if the edge of the cube is 5 inches, we would have: V = (5 in)3. 5 in * 5 in * 5 in = 125 in3, which is the volume of the cube.
Ensure that the Unit of Measurement is Written in Cubic Units (Unit Cubed). In the example above, the edge of the cube was measured in inches, so the volume will be in cubic inches. If the edge of the cube is 3 cm, the volume of the cube will be V = (3 cm)3, or V = 27 cm3.
Calculating the Volume of a Rectangular Prism
Identifying a Rectangular Prism. A rectangular prism, also known as a rectangular parallelepiped, is a three-dimensional shape with six faces, all of which are rectangles. In other words, it’s a three-dimensional rectangle, or simply a box.
- A cube is a special case of a rectangular prism where all the edges are equal.
Determining the Formula. The formula to calculate the volume of a rectangular prism is: Volume = length (denoted as: l) * width (denoted as: w) * height (denoted as: h), or V = lwh.
Finding the Length of the Rectangular Prism. The length is the longest edge of the face of the rectangular prism that lies parallel to the surface on which the shape rests. This length may be specified in a diagram, in the problem statement, or you may need to measure it with a ruler.
- For example, if the length of the rectangular prism is 4 inches, then l = 4 in.
- However, don’t worry too much about distinguishing between length, width, or height. Once you measure the three distinct edge lengths of the rectangular prism, the final result will be the same regardless of the order you assign to them.
Finding the Width of the Rectangular Prism. The width of the rectangular prism is the shorter edge of the face parallel to the surface on which the shape rests. You can determine this value by referring to a diagram, if provided, or by measuring with a ruler.
- For example: If the width of the rectangular prism is 3 inches, then w = 3 in.
- If you measure the edges of the rectangular prism using a ruler or tape measure, make sure to use the same unit of measurement for all measurements. Don't measure one edge in inches and another in centimeters; all measurements should be in the same unit!
Finding the Height of the Rectangular Prism. The height is the distance from the surface on which the shape rests (the bottom face) to the top face of the rectangular prism. You can refer to a given diagram or measure this value with a ruler.
- For example: If the height of the rectangular prism is 6 inches, then h = 6 in.
Substitute the values you found into the formula for the volume of a rectangular box: V = lwh.
- From the examples above, we have: l = 4 in, w = 3 in, h = 6 in. Therefore, V = 4 * 3 * 6, which equals 72.
Make sure to report the result in cubic units (with the unit raised to the power of 3). Since the sides of the rectangular box are measured in inches, the volume of this box should be written as 72 in3.
- If the dimensions of the rectangular box are: l = 2 cm, w = 4 cm, and h = 8 cm, the volume will be: 2 cm * 4 cm * 8 cm, or 64 cm3.
Calculating the Volume of a Circular Cylinder
Identifying a cylinder. A cylinder is a 3D shape with two flat circular bases that are identical, and a curved surface that connects the two bases.
- An AA or AAA battery typically has a cylindrical shape.
Formula for calculating the volume of a circular cylinder. To calculate the volume of a circular cylinder, you need to know its height and the diameter of its base (or the distance from the center to the edge of the circular base). The formula for the volume of a circular cylinder is: V = πr2h, where V is the volume, r is the radius of the base, h is the height of the cylinder, and π is the constant pi.
- In some geometry problems, the answer might be expressed as a ratio of pi, but in most cases, you can round and use the value of pi as 3.14. Be sure to check with your teacher on which version to use.
- The formula for the volume of a circular cylinder is quite similar to that of a rectangular box: multiply the height (h) by the area of the base. For a rectangular box, the base area is l * w, while for a circular cylinder, the area of the base is πr2.
Find the radius of the base. If this value is given on a diagram, you can use it directly. If the problem provides the diameter (usually denoted as d) of the base, simply divide this value by 2 to obtain the radius (since d = 2r).
Measure the cylinder to find the radius of the base. It's important to note that obtaining an accurate measurement of a circle requires precision. One method is to find and measure the widest part of the base of the cylinder and divide this value by 2 to get the radius.
- Another way to calculate the radius is by measuring the circumference of the base (the length of the perimeter of the circle) using a tape measure or a piece of string that can be marked, and then measuring it again with a ruler. Once you have the circumference, use the following formula: C (Circumference) = 2πr. Divide the circumference by 2π (or 6.28), and you will get the radius.
- For example, if the measured circumference is 8 inches, the radius will be 1.27 in.
- If you want to get the most accurate result for the circumference, you can apply and compare the results from both methods above. If there's a significant difference, check again. The method using the circumference usually gives a more precise result.
Calculate the area of the base of the circular cylinder. Substitute the radius into the formula πr2. Then, multiply the radius by itself once more, and multiply the result by π. For example:
- If the radius of the circle is 4 inches (equivalent to 10.16 cm), the area of the base will be A = π42.
- 42 = 4 * 4, or 16. 16 * π (3.14) = 50.24 in2
- If the diameter of the base is known, remember the formula: d = 2r. Simply divide the diameter by 2 to get the radius.
Find the height of the circular cylinder. The height of the cylinder is the distance between the two bases. Look for the height symbol (usually h) on the diagram, or measure it directly with a ruler.
Multiply the area of the base by the height to get the volume. Alternatively, you can use a shortcut by plugging the values for the base radius and height of the cylinder into the formula V = πr2h. For the example above, the radius of the base is 4 inches and the height is 10 inches:
- V = π4210
- π42 = 50.24
- 50.24 * 10 = 502.4
- V = 502.4
The calculated results should be presented in cubic units (cubed unit of measurement). The circular cylinder in the example above is measured in inches, so the volume of this cylinder is in cubic inches: V = 502.4in3. If your circular cylinder is measured in centimeters, the volume should be given in cubic centimeters (cm3).
Calculating the Volume of a Pyramid
Identifying a Pyramid. A pyramid is a three-dimensional shape with a base that is a polygon, and the lateral faces of the pyramid intersect at a point called the apex. A regular polygon pyramid has a base that is a regular polygon, meaning all the edges of the polygon are equal, and all angles are identical.
- We often imagine a pyramid with a square base, where the lateral faces meet at one point, but the base of a pyramid can have 5, 6, or even 100 sides!
- A pyramid with a circular base is called a cone, and we will discuss the volume of a cone later.
Formula for the Volume of a Regular Polygonal Pyramid. The formula to calculate the volume of a regular polygonal pyramid is V = 1/3bh, where b is the area of the base (the polygonal base), and h is the height of the pyramid, which is the distance from the apex to the base.
- The formula for the volume of a regular pyramid is similar to the above, except the projection of the apex of the polygon onto the base is the center of the base. For a slanted pyramid, the projection of the apex onto the base does not coincide with the center of the base.
Calculating the Area of the Base. The formula for calculating the area of the base depends on the number of sides of the polygon that forms the base. For the pyramid shown here, the base is a square with sides measuring 6 inches. The formula for the area of a square is A = s2, where s is the length of the side of the square. Therefore, for this pyramid, the area of the base is (6 in) 2, or 36 in2.
- The formula for the area of a pyramid with a triangular base is: A = 1/2bh, where b is the area of the base and h is the height.
- The area of any polygon can be calculated using the formula A = 1/2pa, where A is the area, p is the perimeter, and a is the apothem, which is the distance from the center of the polygon to the midpoint of any side. This formula is beyond the scope of this lesson, but you can learn more about how to apply this formula in how to calculate the area of a polygon.
Find the Height of the Pyramid. In most cases, this value will be provided in a diagram. For the example we are considering, the height of the pyramid is 10 inches. Image:Calculate Volume Step 30.jpg|center]]
Multiply the area of the base by the height, then divide the result by 3. The formula for calculating the volume of a pyramid is V = 1/3bh. For the pyramid example we are using, the base area is 36 and the height is 10, so the volume is: 36 * 10 * 1/3, or 120.
- If we have another pyramid with a pentagonal base whose area is 26 and height is 8, the volume of this pyramid will be 1/3 * 26 * 8 = 69.33.
Remember to express the result in cubic units. The pyramid we are considering is measured in inches, so the volume of the pyramid will be in cubic inches, 120 in3. If the pyramid's dimensions were measured in meters, the volume would be expressed in cubic meters (m3).
Calculating the Volume of a Cone
Characteristics of a Cone. A cone is a three-dimensional geometric shape with a circular base and a single apex. You can imagine a cone as a pyramid with a circular base.
- If the projection of the apex onto the base coincides with the center of the base, we call it a "right cone." Otherwise, it is called a "oblique cone." However, the formula for the volume of both types of cones is the same.
Formula for the Volume of a Cone. V = 1/3πr2h is the formula to calculate the volume of any cone, where r is the radius of the base, h is the height of the cone, and π is the mathematical constant pi, which is typically approximated as 3.14.
- In the above formula, πr2 represents the area of the base. This shows that the formula for the volume of a cone is essentially the same as the pyramid volume formula, 1/3bh, as we discussed earlier.
Calculating the Area of the Cone's Base. To calculate this value, you need to know the radius of the base, which may be given in the diagram. If the diameter is provided instead of the radius, simply divide the diameter by 2, as the diameter is twice the radius. Then, substitute the radius into the circle area formula, A = πr2.
- For the example provided in the diagram, the radius of the cone's base is 3 inches. Substituting this into the formula gives: A = π(32).
- 32 = 3 * 3, or 9, so A = 9π.
- A = 28.27 in2
Finding the Height of the Cone. The height of the cone is the distance from the apex to the base. In the example we're working with, the height of the cone is 5 inches.
Multiply the Area of the Base by the Height of the Cone. In this example, the area of the cone's base is 28.27 in2 and the height is 5 in, so bh = 28.27 * 5 = 141.35.
To find the volume of the cone, multiply the result from the previous step by 1/3 (or divide by 3). In the previous step, we calculated the volume of a cylinder that could be formed by extending the cone's side and creating a new base. To get the volume of the cone, divide the previous result by 3.
- Thus, in this example, the volume of the cone is 141.35 * 1/3 = 47.12.
- You can simplify the calculations to 1/3π(32)5 = 47.12.
Remember to include the units of volume, such as cubic inches or cubic meters, etc. In the example above, the values are calculated using inches, so the volume should be recorded as 47.12 in3.
Calculating the Volume of a Sphere
Identifying a Sphere. A sphere is a perfectly round three-dimensional object, where the distance from any point on the surface of the sphere to its center is constant. In simpler terms, a sphere is like a ball.
Formula for the Volume of a Sphere. The formula for the volume of a sphere is V = 4/3πr3 (which is “four times pi divided by 3 times the radius cubed”), where r is the radius of the sphere and π is the mathematical constant pi (3.14).
Finding the Radius of a Sphere. If the radius is provided in the diagram, simply locate where it is marked. If the diameter is given, divide it by 2 to find the radius. For example, the radius of the sphere shown in the diagram is 3 inches.
Measure the radius if the value is unknown. If you need to measure a sphere (such as a tennis ball) to find the radius, first get a piece of string long enough to wrap around the sphere. Then, wrap the string around the widest part of the sphere and mark where the string intersects. Use a ruler to measure the length of the string to find the circumference. Divide this value by 2π, or 6.28, to find the radius of the sphere.
- For example, if you measure a ball and get a circumference of 18 inches, divide that by 6.28 to get a radius of 2.87 inches.
- Measuring a sphere might require some skill, so to get the most accurate result, repeat the measurement three times and then take the average (add up the results from the three measurements and divide by 3).
- For instance, if the circumferences measured over three trials were 18 inches, 17.75 inches, and 18.2 inches, add them up (18 + 17.75 + 18.2 = 53.95) and divide by 3 (53.95/3 = 17.98). Use this value to calculate the volume.
Cube the radius to find r3. To cube the radius, multiply the radius by itself three times, so r3 = r * r * r. In our example, if r = 3, then r3 = 3 * 3 * 3, which equals 27.
Multiply the result by 4/3. You can use a calculator, or multiply manually and simplify the resulting fraction. In this case, multiplying 27 by 4/3 gives 108/3, which simplifies to 36.
Multiply the result from the previous step by π to find the volume of the sphere. The final step in calculating the volume of the sphere is to multiply the result from the previous step by π. Round the value of π to two decimal places, which is generally accepted in most math problems (unless your teacher specifies otherwise), and multiply by 3.14 to get the volume of the sphere.
- For example, in our case, 36 * 3.14 = 113.04.
Record the result in cubic units. Since the radius in our example is measured in inches, the volume is expressed in cubic inches, so our result is V = 113.04 cubic inches (113.04 in3).
