6 Ways to Calculate Volume

2024 Author: Samantha Chapman | [email protected]. Last modified: 2024-01-15 13:47

The volume of a solid is the value of how much three-dimensional space the object occupies. You can think of the volume as the amount of water (or sand, or air and so on) that the object can contain once it is completely filled. The most common units of measurement are cubic centimeters (cm³) and cubic meters (m³); in the Anglo-Saxon system instead cubic inches are preferred (in³) and cubic feet (ft³). This article will teach you to calculate the volume of six different solid figures commonly found in math problems (such as cones, cubes and spheres). You will notice that many formulas in the volume are similar to each other, which makes them easy to memorize. Test yourself and see if you can recognize them while reading!

In Brief: Calculate the Volume of Common Figures

1. In a cube or a rectangle parallelepiped you have to measure the height, width and depth and then multiply them together to find the volume. See the details and images.
2. Measure the height of a cylinder and the radius of the base. Use these values and calculate πr², then multiply the result by the height. See details and pictures.
3. The volume of a regular pyramid is equal to ⅓ x base area x height. See details and pictures.
4. The volume of a cone is calculated with the formula: ⅓πr²h, where r is the radius of the base and h the height of the cone. See details and pictures.

5. To find the volume of a sphere, all you need to know is the radius r. Enter its value in the formula ⁴/₃πr³. See details and pictures.

   Steps

   Method 1 of 6: Calculate the Volume of a Cube

   

   Step 1. Recognize a cube

   It is a three-dimensional geometric figure with six equal square faces. In other words, it is a box with all sides equal.

   A six-sided die is a good example of a cube you can find around the house. Sugar cubes and children's wooden blocks with letters are also usually cubes

   

   Step 2. Learn the formula for the volume of the cube

   Since all sides are the same, the formula is very simple. It is V = s³, where V stands for volume and s is the length of one side of the cube.

   To find s³, simply multiplies s three times by itself: s³ = s * s * s.

   

   Step 3. Find the length of one side

   Depending on the type of problem you are given, you may already have this data or you will need to measure it with a ruler. Remember that since all sides are the same in the cube, it doesn't matter which one you consider.

   If you are not 100% sure that the figure in question is a cube, measure each side to make sure they are all the same. If not, you will need to use the method described below to calculate the volume of a rectangular box

   

   Step 4. Enter the side value in the formula V = s³ and do the calculations.

   For example, if you found the side length of the cube to be 5cm, then you should rewrite the formula as follows: V = (5cm)³. 5cm * 5cm * 5cm = 125cm³, that is, the volume of the cube!

   

   Step 5. Remember to express your answer in cubic units

   In the example above, the length of the side of the cube was measured in centimeters, so the volume must be expressed in cubic centimeters. If the side value had been 3 cm, the volume would have been V = (3 cm)³ therefore V = 27 cm³.

   Method 2 of 6: Calculate the Volume of a Rectangle Block

   

   Step 1. Recognize a rectangle box

   This three-dimensional figure, also called a rectangular prism, has six rectangular faces. In other words, it is a "box" with sides that are rectangles.

   A cube is actually a particular rectangle parallelepiped in which all edges are equal

   

   Step 2. Learn the formula for calculating the volume of this figure

   The formula is: Volume = length * depth * height or V = lph.

   

   Step 3. Find the length of the solid

   This is the longest side of the face parallel to the ground (or the one on which the parallelepiped rests). The length can be given by the problem or it needs to be measured with a ruler (or tape measure).

   • For example: the length of this rectangular solid is 4 cm, so l = 4 cm.
   • Don't worry too much about which side you consider like length, depth and height. As long as you measure three different dimensions, the result does not change, regardless of the position of the factors.

   

   Step 4. Find the depth of the solid

   This consists of the shorter side of the face parallel to the ground, the one on which the parallelepiped rests. Again, check if the problem provides this data, or measure it with a ruler or tape measure.

   • Example: the depth of this rectangular parallelepiped is 3 cm so p = 3 cm.
   • If you are measuring the rectangular solid with a meter or a ruler, remember to write down the unit of measurement next to the numerical value and that this is constant for each measurement. Do not measure one side in centimeters and the other in millimeters, always use the same unit!

   

   Step 5. Find the height of the parallelepiped

   This is the distance between the face resting on the ground (or the one on which the solid rests) and the upper face. Locate this information in the problem or find it by measuring the solid with a ruler or tape measure.

   Example: the height of this solid is 6 cm, so h = 6 cm

   

   Step 6. Enter the dimensions of the rectangle box into the formula and do the calculations

   Remember that V = lph.

   In our example, l = 4, p = 3 and h = 6. So V = 4 * 3 * 6 = 72

   

   Step 7. Verify that you have expressed the value in cubic units

   Since the dimensions of the parallelepiped under consideration were measured in centimeters, your answer will be written as 72 cubic centimeters or 72 cm³.

   If the dimensions were: length = 2cm, depth = 4cm and height = 8cm, the volume would have been 2cm * 4cm * 8cm = 64cm³.

   Method 3 of 6: Calculate the Volume of a Cylinder

   

   Step 1. Learn to recognize a cylinder

   It is a solid geometric figure with two identical circular and flat bases with a single curved face that connects them.

   A good example of a cylinder is AA or AAA type batteries

   

   Step 2. Memorize the cylinder volume formula

   To calculate this data, you need to know the height of the figure and the radius of the circular base (the distance between the center and the circumference). The formula is: V = πr²h, where V is the volume, r is the radius of the circular base, h is the height of the solid and π is the constant pi.

   • In some geometry problems the solution can be expressed in terms of pi, but in most cases you can round the constant to 3, 14. Ask your teacher what he prefers.
   • The formula for finding the volume of a cylinder is very similar to that of the rectangular parallelepiped: you simply multiply the height of the solid by the area of the base. In a rectangular parallelepiped the surface of the base is equal to l * p while for the cylinder it is πr², that is, the area of a circle with radius r.

   

   Step 3. Find the radius of the base

   If this value is provided by the problem, simply use the number that is given. If the diameter instead of the radius is disclosed, divide the value by two (d = 2r).

   

   Step 4. Measure the solid, if you don't know its radius

   Be careful because getting accurate readings from a circular object isn't always easy. One solution would be to measure the top face of the cylinder with a ruler or tape measure. Do your best to line up with the widest part of the circle (the diameter) and then divide the figure you get by 2, so you get the radius.

   • Alternatively, measure the circumference of the cylinder (the perimeter) using a tape measure or piece of string on which you can mark the circumference measurement (and then check it with a ruler). Enter the data found in the formula for the circumference: C (circumference) = 2πr. Divide the circumference by 2π (6, 28) and you get the radius.
   • For example, if the circumference you measured is 8cm, then the radius will be 1.27cm.
   • If you need accurate data, you can use both methods to make sure you get similar values. If not, repeat the process. Calculating the radius from the circumference value usually gives more accurate results.

   

   Step 5. Calculate the area of the base circle

   Enter the radius value in the area formula: πr². First multiply the radius once by itself and multiply the product by π. Eg:

   • If the radius of the circle is 4 cm, then the area of the base is A = π4².
   • 4² = 4 * 4 = 16. 16 * π (3, 14) = 50, 24 cm².
   • If you have been given the diameter of the base instead of the radius, remember that this is equal to d = 2r. You will simply have to divide the diameter in half to get the radius.

   

   Step 6. Find the height of the cylinder

   This is the distance between the two circular bases. Find this in the problem or measure it with a ruler or tape measure.

   

   Step 7. Multiply the value of the base area by that of the height of the cylinder and you will get the volume

   Or you can avoid this step by entering the dimensions of the solid directly into the formula V = πr²h. In our example, the cylinder with a radius of 4 cm and a height of 10 cm will have a volume of:

   • V = π4²10
   • π4² = 50, 24
   • 50, 24 * 10 = 502, 4
   • V = 502.4

   

   Step 8. Remember to express the result in cubic units

   In our example, the dimensions of the cylinder were measured in centimeters, so the volume must be expressed in cubic centimeters: V = 502, 4 cm³. If the cylinder had been measured in millimeters, the volume would have been indicated in cubic millimeters (mm³).

   Method 4 of 6: Calculate the Volume of a Regular Pyramid

   

   Step 1. Understand what a regular pyramid is

   It is a solid figure with a base polygon and the side faces that join at a vertex (the tip of the pyramid). A regular pyramid is based on a regular polygon (with all sides and angles equal).

   • Most of the time we imagine a square-based pyramid with sides converging at a single point, but there are pyramids with a base of 5, 6 and even 100 sides!
   • A pyramid with a circular base is called a cone and will be discussed later.

   

   Step 2. Learn the volume formula of a regular pyramid

   This is V = 1 / 3bh, where b is the area of the base of the pyramid (the polygon located at the bottom of the solid) and h is the height of the pyramid (the vertical distance between the base and the vertex).

   The volume formula is valid for all types of straight pyramids, where the vertex is perpendicular to the center of the base, and for oblique ones, where the vertex is not centered

   

   Step 3. Calculate the area of the base

   The formula depends on how many sides the geometric figure serving as a base has. The one in our diagram has a square base with 6 cm sides. Remember that the formula for the area of the square is A = s² where s is the length of the side. In our case, the base area is (6 cm) ² = 36 cm².

   • The formula for the area of the triangle is: A = 1 / 2bh, where b is the base of the triangle and h its 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 and a is the apothem, the distance between the center of the geometric figure and the midpoint of any side. This is a rather complex calculation that is beyond the scope of this article, however you can read this article where you will find valid instructions. Alternatively, you can find "shortcuts" online with automatic polygon area calculators.

   

   Step 4. Find the height of the pyramid

   In most cases this data is indicated in the problem. In our specific example, the pyramid has a height of 10 cm.

   

   Step 5. Multiply the area of the base by its height and divide the result by 3, in this way you get the volume

   Remember that the volume formula is: V = 1 / 3bh. In the pyramid of the example with base 36 and height 10, the volume is: 36 * 10 * 1/3 = 120.

   If we had had a different pyramid, with a pentagonal base of area 26 and height 8, the volume would have been: 1/3 * 26 * 8 = 69.33

   

   Step 6. Remember to express the result in cubic units

   The dimensions of our pyramid have been indicated in centimeters, so the volume must be expressed in cubic centimeters: 120 cm³. If the pyramid had been measured in meters, the volume would be expressed in cubic meters (m³).

   Method 5 of 6: Calculate the Volume of a Cone

   

   Step 1. Learn the properties of the cone

   It is a three-dimensional solid with a circular base and a single vertex (the tip of the cone). An alternative way to think of the cone is to think of it as a special pyramid with a circular base.

   If the vertex of the cone is perpendicular to the center of the circle of the base, it is called a "right cone". If the vertex is not centered with the base, it is called an "oblique cone". Thankfully, the volume formula is the same, whether it's an oblique or a straight cone

   

   Step 2. Learn the cone volume formula

   This is: V = 1 / 3πr²h, where r is the radius of the circular base, h the height of the cone and π is the constant pi which can be approximated to 3, 14.

   The part of the formula πr² refers to the area of the circular base of the cone. For this, you can think of it as the general formula for the volume of a pyramid (see the previous method) which is V = 1 / 3bh!

   

   Step 3. Calculate the area of the circular base

   To do this, you need to know its radius, which should be indicated in the problem data or in the diagram. If you are given the diameter, remember that you just have to divide it by 2 to find the radius (since d = 2r). At this point enter the value of the radius in the formula A = πr² and find the base area.

   • In the example of our diagram, the radius of the base is 3 cm. When you insert this data into the formula you get: A = π3².
   • 3² = 3 * 3 = 9 so A = 9π.
   • A = 28.27 cm²

   

   Step 4. Find the height of the cone

   This is the vertical distance between the vertex and the base of the solid. In our example, the cone has a height of 5 cm.

   

   Step 5. Multiply the height of the cone by the area of the base

   In our case, the area is 28, 27 cm² and the height is 5 cm, so bh = 28, 27 * 5 = 141, 35.

   

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

   In the previous step we practically calculated the volume of a cylinder with the walls extending upwards, perpendicular to the base; however, since we are considering a cone whose walls converge towards the vertex, we must divide this value by 3.

   • In our case: 141, 35 * 1/3 = 47, 12 that is the volume of the cone.
   • To reiterate the concept: 1 / 3π3²5 = 47, 12.

   

   Step 7. Remember to express your answer in cubic units

   Since our cone was measured in centimeters, its volume must be expressed in cubic centimeters: 47, 12 cm³.

   Method 6 of 6: Calculate the Volume of a Sphere

   

   Step 1. Recognize a sphere

   It is a perfectly round three-dimensional object where every point on the surface is equidistant from the center. In other words, a sphere is a ball-shaped object.

   

   Step 2. Learn the formula for calculating the volume of the sphere

   This is: V = 4 / 3πr³ (pronounced "four thirds pi r and r cubed"), where r stands for 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 indicated in the diagram, then it is not difficult to find it. If you are given the diameter data, you need to divide this value by 2 and you will find the radius. For example, the radius of the sphere in the diagram is 3 cm.

   

   Step 4. Measure the sphere if the radius data is not indicated

   If you need to measure a spherical object (such as a tennis ball) to find the radius, first you need to get a string long enough to be wrapped around the object. Next, wrap the string around the sphere at its widest point (or equator) and make a mark where the string overlaps itself. Then measure the segment of string with a ruler and get the circumference value. Divide this number by 2π, or 6, 28, and you get the radius of the sphere.

   • Let's consider the example in which the circumference of the tennis ball is 18 cm: divide this number by 6, 28 and you get a value for the radius of 2.87 cm.
   • It is not easy to measure a spherical object, the best thing is to take three measurements and calculate the average (add the values together and divide the result by 3), in this way you will get the most accurate data possible.
   • For example, suppose the three tennis ball circumference measurements are: 18cm, 17, 75cm, and 18.2cm. You should add these numbers together (18 + 17, 75 + 18, 2 = 53, 95) and then divide the result by 3 (53, 95/3 = 17, 98). Use this average value for volume calculations.

   

   Step 5. Cub the radius to find the value of r³.

   This simply means multiplying the data three times by itself, so: r³ = r * r * r. Always following the logic of our example, we have that r = 3, hence r³ = 3 * 3 * 3 = 27.

   

   Step 6. Now multiply the result by 4/3

   You can use a calculator or do the multiplication by hand and then simplify the fraction. In the example of the tennis ball we will have that: 27 * 4/3 = 108/3 = 36.

   

   Step 7. At this point multiply the obtained value by π and you will find the volume of the sphere

   The last step involves multiplying the result found so far by the constant π. In most math problems, this is rounded to the first two decimal places (unless your teacher gives different instructions); so you can easily multiply by 3, 14 and find the final solution to the question.

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

   

   Step 8. Express your answer in cubic units

   In our example we have expressed the radius in centimeters, so the volume value will be V = 113.09 cubic centimeters (113.09 cm³).