Calculate Volume

The volume of a shape is the measure of how much three-dimensional space that shape takes up.[1] You can also think of the volume of a shape as how much water (or air, or sand, etc.) the shape could hold if it was filled completely. Common units of volume include cubic centimeters (cm3), cubic meters (m3), cubic inches (in3), and cubic feet (ft3).[2] This article will teach you how to calculate the volume of six different three-dimensional shapes that are commonly found on math tests, including cubes, spheres, and cones. You might notice that a lot of the volume formulas share similarities that can make them easier to remember. See if you can spot them along the way!

Steps

Calculating the Volume of a Cube

  1. Recognize a cube. A cube is a three-dimensional shape that has six identical square faces.[3] In other words, it is a box shape with equal sides all around.
    • A 6-sided die is a good example of a cube you might find in your house. Sugar cubes, and children's letter blocks are also usually cubes.
  2. Learn the formula for the volume of a cube. Since all of the side lengths of a cube are the same, the formula for the volume of a cube is really easy. It is V = s3 where V stands for volume, and s is the length of the sides of the cube.
    • To find s3, simply multiply a by itself 3 times: s3 = s * s * s
  3. Find the length of one side of the cube. Depending on your assignment, the cube will either be labeled with this information, or you may have to measure the side length with a ruler. Remember that since it is a cube, all of the side lengths should be equal so it doesn't matter which one you measure.
    • If you are not 100% sure that your shape is a cube, measure each of the sides to determine if they are equal. If they are not, you will need to use the method below for Calculating the Volume of a Rectangular Solid.
  4. Plug the side length into the formula V = s3 and calculate. For example, if you find that the length of the sides of your cube is 5 inches, then you should write the formula out as follows: V = (5 in)3. 5 in * 5 in * 5 in = 125 in3, the volume of our cube!
  5. Be sure to state your answer in cubic units. In the above example, the side length of our cube was measured in inches, so the volume was given in cubic inches. If the side length of the cube had been 3 centimeters, for example, the volume would be V = (3 cm)3, or V = 27cm3.

Calculating the Volume of a Rectangular Prism

  1. Recognize a rectangular solid. A rectangular solid, also known as a rectangular prism, is a three-dimensional shape with six sides that are all rectangles.[4] In other words, a rectangular solid is simply a three-dimensional rectangle, or box shape.
    • A cube is really just a special rectangular solid in which the sides of all of the rectangles are equal.
  2. Learn the formula for calculating the volume of a rectangular solid. The formula for the volume of a rectangular solid is Volume = length * width * height, or V = lwh. [5]
  3. Find the length of the rectangular solid. The length is the longest side of the rectangular solid that is parallel to the ground or surface it is resting on. The length may be given in a diagram, or you may need to measure it with a ruler or tape measure.
    • Example: The length of this rectangular solid is 4 inches, so l = 4 in.
    • Don't worry too much about which side is the length, which is the width, etc. As long as you end up with three different measurements, the math will come out the same regardless of how your arrange the terms.
  4. Find the width of the rectangular solid. The width of the rectangular solid is the measurement of the shorter side of the solid, parallel to the ground or surface the shape is resting on. Again, look for a label on the diagram indicating the width, or measure your shape with a ruler or tape measure.
    • Example: The width of this rectangular solid is 3 inches, so w = 3 in.
    • If you are measuring the rectangular solid with a ruler or tape measure, remember to take and record all measurements in the same units. Don't measure one side in inches another in centimeters; all measurements must use the same unit!
  5. Find the height of the rectangular solid. This height is the distance from the ground or surface the rectangular solid is resting on to the top of the rectangular solid. Locate the information in your diagram, or measure the height using a ruler or tape measure.
    • Example: The height of this rectangular solid is 6 inches, so h = 6 in.
  6. Plug the dimensions of the rectangular solid into the volume formula and calculate. Remember that V = lwh.
    • In our example, l = 4, w = 3, and h = 6. Therefore V = 4 * 3 * 6, or 72.
  7. Be sure to express your answer in cubic units. Since our example rectangle was measured in inches, the volume should be written as 72 cubic inches, or 72 in3.
    • If the measurements of our rectangular solid were: length = 2 cm, width = 4 cm, and height = 8 cm, the Volume would be 2 cm * 4 cm * 8 cm, or 64cm3.

Calculating the Volume of a Cylinder

  1. Learn to identify a cylinder. A cylinder is a three-dimensional shape that has two identical flat ends that are circular in shape, and a single curved side that connects them.[6]
    • A can is a good example of a cylinder, so is a AA or AAA battery.
  2. Memorize the formula for the volume of a cylinder. To calculate the volume of a cylinder, you must know its height and the radius of the circular base (the distance from the center of the circle to its edge) at the top and bottom. The formula is V = πr2h, where V is the Volume, r is the radius of the circular base, h is the height, and π is the constant pi.
    • In some geometry problems the answer will be given in terms of pi, but in most cases it is sufficient to round pi to 3.14. Check with your instructor to find out what she would prefer.
    • The formula for finding the volume of a cylinder is actually very similar to that for a rectangular solid: you are simply multiplying the height of the shape by the surface area of its base. In a rectangular solid, that surface area is l * w, for the cylinder it is πr2, the area of a circle with radius r.
  3. Find the radius of the base. If it is given in the diagram, simply use that number. If the diameter is given instead of the radius, you simply need to divide the value by 2 to get the radius (d = 2r).
  4. Measure the object if the radius is not given. Be aware that getting precise measurement of a circular solid can be a bit tricky. One option is to measure the base of the cylinder across the top with a ruler or tape measure. Do your best to measure the width of the cylinder at its widest part, and divide that measurement by 2 to find the radius.
    • Another option is to measure the circumference of the cylinder (the distance around it) using a tape measure or a length of string that you can mark and then measure with a ruler. Then plug the measurement into the formula: C (circumference) = 2πr. Divide the circumference by 2π (6.28) and that will give you the radius.
    • For example, if the circumference you measured was 8 inches, the radius would be 1.27in.
    • If you need a really precise measurement, you might use both methods to make sure that your measurements are similar. If they are not, double check them. The circumference method will usually yield more accurate results.
  5. Calculate the area of the circular base. Plug the radius of the base into the formula πr2. Then multiply the radius by itself one time, and then multiply the product by π. For example:
    • If the radius of the circle is equal to 4 inches, 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 given instead of the radius, remember that d = 2r. You simply need to divide the diameter in half to find the radius.
  6. Find the height of the cylinder. This is simply the distance between the two circular bases, or the distance from the surface the cylinder is resting on to its top. Find the label in your diagram that indicates the height of the cylinder, or measure the height with a ruler or tape measure.
  7. Multiply the area of the base times the height of the cylinder to find the volume. Or you can save a step and simply plug the values for the cylinder's dimensions into the formula V = πr2h. For our example cylinder with radius 4 inches and height 10 inches:
    • V = π4210
    • π42 = 50.24
    • 50.24 * 10 = 502.4
    • V = 502.4
  8. Remember to state your answer in cubic units. Our example cylinder was measured in inches, so the volume must be expressed in cubic inches: V = 502.4in3. If our cylinder had been measured in centimeters, the volume would be expressed in cubic centimeters (cm3).

Calculating the Volume of a Regular Square Pyramid

  1. Understand what a regular pyramid is. A pyramid is a three-dimensional shape with a polygon for a base, and lateral faces that taper at an apex (the point of the pyramid).[7] A regular pyramid is a pyramid in which the base of the pyramid is a regular polygon, meaning that all of the sides of the polygon are equal in length, and all of the angles are equal in measure.[8]
    • We most commonly imagine a pyramid as having a square base, and sides that taper up to a single point, but the base of a pyramid can actually 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.
  2. Learn the formula for the volume of a regular pyramid. The formula for the volume of a regular pyramid is V = 1/3bh, where b is the area of the base of the pyramid (the polygon at the bottom) and h is the height of the pyramid, or the vertical distance from the base to the apex (point).
    • The volume formula is the same for right pyramids, in which the apex is directly above the center of the base, and for oblique pyramids, in which the apex is not centered.
  3. Calculate the area of the base. The formula for this will depend on the number of sides the base of the pyramid has. In the pyramid in our diagram, the base is a square with sides that are 6 inches in length. Remember that the formula for the area of a square is A = s2 where s is the length of the sides. So for this pyramid, the area of the base is (6 in) 2, or 36in2.
    • 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.
    • It is possible to 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 apothem, or distance from the center of the shape to the midpoint of any of its sides. This is a pretty involved calculation that goes beyond the scope of this article, but check out Calculate-the-Area-of-a-Polygon for some great instructions on how to use it. Or you can make your life easy and search for a Regular Polygon Calculator online.[9]
  4. Find the height of the pyramid. In most cases, this will be indicated in the diagram. In our example, the height of the pyramid is 10 inches.
  5. Multiply the area of the base of the pyramid by its height, and divide by 3 to find the volume. Remember that the formula for the volume is V = 1/3bh. In our example pyramid, that had a base with area 36 and height 10, the volume is: 36 * 10 * 1/3, or 120.
    • If we had a different pyramid, with a pentagonal base with area 26, and height of 8, the volume would be: 1/3 * 26 * 8 = 69.33.
  6. Remember to express your answer in cubic units. The measurements of our example pyramid were given in inches, so its volume must be expressed in cubic inches, 120in. If our pyramid had been measured in meters, the volume would be expressed in cubic meters (m3) instead.3

Calculating the Volume of a Cone

  1. Learn the properties of a cone. A cone is a 3-dimesional solid that has a circular base and a single vertex (the point of the cone). Another way to think of this is that a cone is a special pyramid that has a circular base.[10]
    • If the vertex of the cone is directly above the center of the circular base, the cone is called a "right cone". If it is not directly over the center, the cone is called an "oblique cone." Fortunately, the formula for calculating the area of a cone is the same whether it is right or oblique.
  2. Know the formula for calculating the volume of a cone. The formula is V = 1/3πr2h, where r is the radius of the circular base of the cone, h is the height of the cone, and π is the constant pi, which can be rounded to 3.14.
    • The πr2 part of the formula refers to the area of the circular base of the cone. The formula for the volume of the cone is thus 1/3bh, just like the formula for the volume of a pyramid in the method above!
  3. Calculate the area of the circular base of the cone. To do this, you need to know the radius of the base, which should be listed in your diagram. If you are instead given the diameter of the circular base, simply divide that number by 2, since the diameter is simply 2 times the radios (d = 2r). Then plug the radius into the formula A = πr2 to calculate the area.
    • In the example in the diagram, the radius of the circular base of the cone is 3 inches. When we plug that into the formula we get: A = π32.
    • 32 = 3 *3, or 0, so A = 9π.
    • A = 28.27in2
  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.
  5. Multiply the height of the cone by the area of the base. In our example, the area of the base is 28.27in2 and the height is 5in, so bh = 28.27 * 5 = 141.35.
  6. Now multiply the result by 1/3 (or simply divide by 3) to find the volume of the cone. In the above step, we actually calculated the volume of the cylinder that would be formed if the walls of the cone extended straight up to another circle, instead of slanting in to a single point. Dividing by 3 gives us the volume of just the cone itself.
    • In our example, 141.35 * 1/3 = 47.12, the volume of our cone.
    • To restate it, 1/3π325 = 47.12
  7. Remember to express your answer in cubic units. Our cone was measured in inches, so its volume must be expressed in cubic inches: 47.12in3.

Calculating the Volume of a Sphere

  1. Spot a sphere. A sphere is a perfectly round three-dimensional object, in which every point on the surface is an equal distance from the center. In other words, a sphere is a ball-shaped object.[11]
  2. Learn the formula for the volume of a sphere. The formula for the volume of a sphere is V = 4/3πr3 (stated: "four-thirds times pi r-cubed") where r is the radius of the sphere, and π is the constant pi (3.14).[12]
  3. Find the radius of the sphere. If the radius is given in the diagram, then finding r is simply a matter of locating it. If the diameter is given, you must divide this number by 2 to find the radius. For example, the radius of the sphere in the diagram is 3 inches.
  4. Measure the sphere if the radius is not given. If you need to measure a spherical object (like a tennis ball) to find the radius, first find a piece of string large enough to wrap around the object. Then wrap the string around the object at its widest point and mark the points where the string overlaps itself. Then measure the string with a ruler to find the circumference. Divide that value by 2π, or 6.28, and that will give you the radius of the sphere.
    • For example, if you measure a ball and find its circumference is 18 inches, divide that number by 6.28 and you will find that the radius is 2.87in.
    • Measuring a spherical object can be a little tricky, so you might want to take 3 different measurements, and then average them together (add the three measurements together, then divide by 3) to make sure you have the most accurate value possible.
    • For example, if your three circumference measurements were 18 inches, 17.75 inches, and 18.2 inches, you would add those three values together (18 + 17.5 + 18.2 = 53.95) and divide that value by 3 (53.95/3 = 17.98). Use this average value in your volume calculations.
  5. Cube the radius to find r3. Cubing a number simply means multiplying the number by itself 3 times, so r3 = r * r * r. In our example, r = 3, so r3 = 3 * 3 * 3, or 27.
  6. Now multiply your answer by 4/3. You can either use your calculator, or do the multiplication by hand and then simplify the fraction. In our example, multiplying 27 by 4/3 = 108/3, or 36.
  7. Multiply the result by π to find the volume of the sphere. The last step in calculating the volume is simply to multiply the result so far by π. Rounding π to two digits is usually sufficient for most math problems (unless your teacher specified otherwise,) so multiply by 3.14 and you have your answer.
    • In our example, 36 * 3.14 = 113.09.
  8. Express your answer in cubic units. In our example, the measurement of the radius of the sphere was in inches, so our answer is actually V = 113.09 cubic inches (113.09 in3).

Sources and Citations

  1. ↑ http://www.mathsisfun.com/definitions/volume.html
  2. ↑ http://www.mathsisfun.com/measure/us-standard-volume.html
  3. ↑ https://www.mathsisfun.com/definitions/cube.html
  4. ↑ http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_3Dprisms.xml
  5. ↑ http://www.studyzone.org/mtestprep/math8/g/rectvolumel.cfm
  6. ↑ https://www.mathsisfun.com/definitions/cylinder.html
  7. ↑ http://www.mathwords.com/p/pyramid.htm
  8. ↑ http://www.mathwords.com/r/regular_pyramid.htm
  9. ↑ http://www.calculatorsoup.com/calculators/geometry-plane/polygon.php
  10. ↑ http://www.mathopenref.com/cone.html
  11. ↑ https://www.mathsisfun.com/definitions/sphere.html
  12. ↑ http://www.aaamath.com/exp79_x8.htm
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