How to Calculate Volume

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The volume of a shape is a value that indicates how much of a three-dimensional space the shape occupies. ^{[1] X Research Source} You can also imagine the volume of a shape as the amount of water (or air, or sand, etc.) that the shape can hold when filled with the above objects. Common units of volume include cubic centimeters (cm ³ ), cubic meters (m ³ ), cubic inches (in ³ ), and cubic feet (ft ³ ). ^{[2] X Research Resources} This article will show you how to calculate the volume of 6 three-dimensional cubes commonly encountered in math tests, including cube, rectangle, cylinder, pyramid, conical and spherical. You may find that the formulas for volume have the same parts that you can use to remember them. Follow the steps below to see if you can spot the similarities!

Table of Contents

Calculating the Volume of a Cube

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Identify the cube. A cube is a three-dimensional cube with six square faces. ^{[3] X Research Source} In other words, this is a box with all sides equal.

A six-sided die is an example of a cube that you might find at home. Sugar tablets or children’s literacy blocks are also often cube-shaped.

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Formula for calculating the volume of a cube. Since all sides of a cube are equal, the formula for calculating the volume of a cube is also very simple. That is: V = s ³ where V is the volume, s is the side of the cube.

To find s ³ , you just need to multiply s by itself 3 times, ie: s ³ = s * s * s

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Find the length of one side of the cube. Depending on the case, the problem may give this value, or you may have to measure the sides of the cube yourself with a ruler. Since this is a cube, which means all sides are equal, you only need to measure any one side.

If you are not 100% sure that the cube you are measuring is a cube, measure all the sides and see if the values are equal. If they are not equal, you need to apply the method of calculating the volume of the rectangular box that will be described in the next section.

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Substitute the measured length into the formula V = s ³ and calculate. For example, if the side of the cube is 5 inches, we get: V = (5 in) ³ . 5 in * 5 in * 5 in = 125 in ³ , this is the volume of the cube.

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Make sure you write the unit of measure in blocks (to the 3 power of the unit of measure). In the example above, the sides of the cube are measured in inches, so the volume will be in cubic inches. If the side of the cube is 3 cm, the volume of the cube will be V = (3 cm) ³ , or V = 27 cm ³ .

Calculating the Volume of a Rectangular

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Identify rectangular boxes. A rectangular prism, also known as a rectangular prism, is a three-dimensional cube with six rectangular faces. ^{[4] X Research Source} In other words, a rectangular box is simply a three-dimensional rectangle, or box.

A cube is a special form of a rectangular box with the sides of the rectangular box being equal.

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Define formula. The formula for calculating the volume of a rectangular box is: Volume = length (symbol: l) * width (symbol: w) * height (symbol: h), or V = lwh. ^{[5] X Research Sources}

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Find the length of the rectangular box. The length is the longest side of the face of the box that is parallel to the plane in which the shape is located. The length can be specified in the diagram, the problem or you must use a ruler to measure.

For example, the length of the rectangular box is 4 inches, so l= 4 in.
However, you do not need to worry too much about determining which is the length, which is the width, and which is the height. When you measure the sides of the rectangular box and you get 3 different values, the final calculation result will be the same no matter how you arrange the elements.

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Find the width of the rectangular box. The width of the rectangular box is the remaining side (which is the shorter side) of the face parallel to the plane in which the box is located. You can determine this value by looking at the chart, if available, or using a ruler to measure it.

Example: The width of the rectangular box is 3 inches, so w = 3 in.
If you are measuring the side of a rectangular box with a ruler or tape measure, be sure to use the same unit of measure for all measurements. Don’t measure one side in inches and the other in centimeters; All measurements need to have the same unit of measure!

Calculating the Volume of a Circular Cylinder

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Identify cylinders. A cylinder is a space cube with two flat bases that are two identical circles and a curved surface connecting the two bases. ^{[6] X Research Source}

An AA or AAA battery is usually round in shape.

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Formula for calculating the volume of a circular cylinder. To calculate the volume of a circular cylinder, you need to know the height of the shape and the diameter of the base (or the distance from the center to the side of the circle). The formula for calculating the volume of a circular cylinder is as follows: V = πr ² h 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 questions, the answer may be given as a ratio of pi, but in most cases we can round off and take the value of pi to be 3.14. Ask your teacher which format you should use.
The formula for calculating the volume of a circular cylinder is very similar to the formula for calculating the volume of a rectangular box: multiply the height (h) by the area of the base. For a rectangular box, the area of the base is l * w, for a circular cylinder, the area of the base of a circle of radius r is πr ² .

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Find the radius of the bottom face. If this value is recorded in the schema, you can use it. If the problem is for the diameter (usually denoted d) of the base, you just need to divide this value by 2 to get the radius (because d = 2r).

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Measure the cylinder to find the radius of the base. It should be noted that to get a certain exact parameter of a circle requires your ingenuity. The first way you can use that is to find and measure the widest part of the base of the circular cylinder and divide that value by 2 to get the radius.

Another way to calculate the radius is to measure the circumference of the base (the length of the circle’s outline) with a tape measure or a piece of string that you can mark, then measure again with a ruler. Once you get the circumference, you apply the following formula: C(Perimeter) = 2πr. Divide the circumference by 2π (or 6.28) and you will find the value of the radius.
For example, if the circumference you measured was 8 inches, the radius would be 1.27 in.
If you want to find the really exact value of the circumference, you can apply and compare the results obtained from the two methods above, if the results are significantly different, check again. The circumferential method will usually give more accurate results.

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Calculate the area of the base of the circle. Substitute the value of the radius into the formula πr ² . Then multiply the radius by itself again, multiplying the result by π. For example:

If the radius of the circle is 4 inches (10.16 cm), the area of the base will be A = π4 ² .
4 ² = 4 * 4, or 16. 16 * (3.14) = 50.24 in ²
If the diameter of the base is known, remember the formula: d = 2r. You just need to divide the value of the diameter by 2 to get the value of the radius.

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Find the height of the circular cylinder. The height of the circular cylinder is the distance between the two bases. Look for the height symbol (usually h) on the diagram or use a ruler to measure it directly.

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Multiply the area of the base by the height to get the volume. Or you can turn it off by substituting the base radius and height of the cylinder into the formula V = πr ² h. For the example above, the base radius is 4 inches and the height is 10 inches:

V = 4 ² 10
4 ² = 50.24
50.24 * 10 = 502.4
V = 502.4

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Calculation results should be expressed in blocks (power 3 of the unit of measure). The cylinder in the above example is measured in inches, so the volume of this circular cylinder is inches to the 3 power: V = 502.4in ³ . If your circular cylinder is measured in centimeters, its volume should be stated in cubic centimeters (cm ³ ).

Calculate the volume of the pyramid

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Identify the pyramid. A pyramid is a spatial shape whose base is a polygon and the sides of the pyramid intersect at a point called the vertex of the pyramid. ^{[7] X Source of Research} A regular polygonal pyramid is a pyramid whose base is a regular polygon, i.e. all sides of the polygon are equal and all angles of the polygon are also equal. ^{[8] X Research Sources}

We often imagine a pyramid with a square base and its faces intersecting at one point, but the base of a pyramid can have 5, 6, or even 100 sides!
A pyramid whose base is a circle is called a cone, we will talk about the volume of the cone later.

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Formula for the volume of a regular polygonal pyramid. The formula for the volume of a regular polygon pyramid is V=1/3bh, where b is the volume of the base (bottom polygon) and h is the height of the pyramid, which is also the distance from the top of the pyramid to the face. its bottom).

The formula for the volume of a regular pyramid is the same as above, in that the projection of the vertex of the polygon to the base is the center of the base, and for an oblique pyramid, the projection of the vertex to the base is not the center of the base. bottom.

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Calculate the area of the bottom surface. The formula for calculating the area of the base depends on the number of sides of the polygon forming the base. For the pyramid in the diagram we have here, the base is a square with sides measuring 6 inches. The formula for calculating the area of a square is A = s ² , where s is the length of the side of the square. So for this pyramid, the area of the base is (6 in) ² , or 36 in ² .

The formula for the volume of a pyramid whose base is a triangle is: A = 1/2bh, where b is the area of the base and h is the height.
We can calculate the area of any polygon by applying the formula A = 1/2pa, where A is the area, p is the perimeter and a is the midpoint, the midpoint is the distance from the center of the polygon. of the polygon to the midpoint of any side. This formula is beyond the scope of this article, but you can also see how to calculate the area of a polygon for a better understanding of how to apply the above formula.

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Find the height of the pyramid. In most cases, this value will be given according to the schema. For the example we are looking at, the height of the pyramid is 10 inches. Image:Calculate Vpume Step 30.jpg|center]]

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Multiply the area of the base by the height, then divide the result by 3. We have the formula for the volume of the pyramid as V=1/3bh. With the pyramid we’re taking as an example, 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 base of a pentagon with area 26, height 8, volume of this pyramid will be 1/3 * 26 * 8 = 69.33.

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Remember to express your results in blocks (to the 3 power of the unit of measure). The size of the pyramid we are considering is measured in inches, so the size of the pyramid will be in cubic inches, 120 in ³ . If the pyramid has dimensions expressed in meters, the volume of the pyramid will be in m ³ .

Calculating the Volume of a Cone

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Features of the cone. A cone is a three-dimensional cube with a circular base and a single top. You can imagine a cone as a pyramid whose base is a circle. ^{[9] X Research Source}

If the projection of the vertex to the base of the cone coincides with the center of the base, it is called a “regular cone”. Otherwise we call it “oblique cone”. However, the formula for the volume of both cones is the same.

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Formula for calculating the volume of a cone. V = 1/3πr ² h 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 constant pi, we can round and get the value. The value of π is 3.14.

In the above formula, πr ² is the area of the base surface. From this we can see that the formula for calculating the volume of the cone is 1/3bh, which is the same formula for calculating the volume of the pyramid that we considered above.

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Calculate the area of the base of the cone. To calculate this value, we need to know the radius of the base, which can be given in the diagram. If the problem gives a diameter instead of a radius, you can simply divide the diameter by 2 because the diameter is 2 times the radius. Then substitute the radius value found in the formula for calculating the area of a circle A = πr ² .

For the example given in the diagram, the radius of the base of the cone is 3 inches. Substituting this value into the formula, we have: A = π3 ² .
3 ² = 3 *3, or 9, so A = 9π.
A = 28.27 in ²

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Find the height of the cone. The height of the cone is the distance between the top of the cone and its base. In the example we are considering, the height of the cone is 5 inches.

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Multiply the area of the base by the height of the cone. In this example, the area of the cone is 28.27 in ² and the height is 5 in, so bh = 28.27 * 5 = 141.35.

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To calculate the volume of the cone, multiply the value obtained in the above calculation by 1/3 (or divide it by 3). In the previous step, we calculated the volume of the cylinder that would form if the side faces of the cone were expanded and formed a different base face instead of clustering at one point. Divide the value obtained in the previous step by 3, we will get the calculation of the cone we are considering.

So, in this example, the volume of the cone is 141.35 * 1/3 = 47.12.
We can reduce the recalculation steps and get 1/3π3 ² 5 = 47.12

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Don’t forget to put the units of the volume in cubic inches or cubic meters, etc. In the example above, the values are in inches, so the volume should be recorded as 47.12 in ³ .

Calculating the Volume of a Sphere

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Sphere recognition. A sphere is a perfectly circular spatial object whose distance from any point on the sphere to the center of the sphere is a constant number. In other words, the sphere is the shape of a ball. ^{[10] X Research Source}

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Formula for calculating the volume of a sphere. The formula for the volume of a sphere is V = 4/3πr ³ (literal: “four times pi divided by 3 times r to the power of 3”) where r is the radius of the sphere, π is the constant pi (3.14). ^{[11] X Research Source}

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Find the radius of the sphere. If the radius is given in the diagram, finding the radius is just seeing where it is marked. If the problem gives a diameter, we find the radius by dividing the diameter in half. For example, the radius of the sphere in the diagram given here is 3 inches.

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Measure the radius if this value is not known. If you need to measure a sphere (like a tennis ball) to find the radius, first find a piece of string long enough to wrap around the sphere. Then use this piece of string to wrap around the sphere at the widest part and mark the intersection of the string. Use a ruler to measure the length of the string to get the circumference. Divide this value by 2π, or 6.28, to get the radius of the sphere.

For example, if you measure a ball and get the circumference of the ball to be 18 inches, divide that number by 6.28 and you get the value of the radius to be 2.87 in.
Measuring a sphere can take some ingenuity, so to get the most accurate results possible, you should measure 3 times and then average the value (add the value obtained after 3 measurements). again and then divide by 3).
For example, if the circumference you get after 3 measurements is 18 inches, 17.75 inches, and 18.2 inches, add these values together (18 + 17.5 + 18.2 = 53.95) ) and divide the total found by 3 (53.95/3 = 17.98). Use this value to calculate the volume.

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The radius cap 3 has to be r ³ . Exposing the radius to 3 is just multiplying the radius by itself 3 times, so r ³ = r * r * r. In the example we’re looking at, r = 3, so r ³ = 3 * 3 * 3, or 27.

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Multiply the result found by 4/3. You can use a calculator, or multiply by hand and then reduce the fraction you find. In the example we are considering, multiplying 27 by 4/3 gives us 108/3, reducing this fraction gives us 36.

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Take the result of the multiplication in the previous step and multiply it by π to calculate the volume of the sphere. The final step in calculating the volume of the sphere is to multiply the result obtained in the previous step by π. Round the value of to 2 decimal places, which is generally accepted in most math problems (unless your teacher tells you otherwise), so multiply by 3.14 and you’ll get the volume. globular.

In the example under consideration, 36 * 3.14 = 113.04.

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Record the results obtained in cubic units. Since in our example we have the radius of the sphere in inches, our result is V = 113.04 cubic inches (113.04 in ³ ).

Steps