Lesson Explainer: Volumes of Spheres | Nagwa Lesson Explainer: Volumes of Spheres | Nagwa

# Lesson Explainer: Volumes of Spheres Mathematics • Second Year of Preparatory School

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In this explainer, we will learn how to find volumes of spheres and solve problems including real-life situations.

Recall that in the formulas for calculating the area and circumference of a circle, we met the special mathematical constant . We pronounce this Greek letter in the same way as the word pie, and it is important to remember that it just represents a number for which we sometimes use the simpler approximations 3.14 or . In view of this, it is perhaps not surprising that we meet again when we move on to the three-dimensional problem of calculating the volume of a sphere.

### Formula: Volume of a Sphere

The volume, , of a sphere of radius is given by the formula .

Since and are just numbers, this means that as long as we know the radius of a sphere, we can always apply this formula to find its volume. Let us start by looking at an example that features a diagram.

### Example 1: Finding the Volume of a Sphere given Its Radius

Work out the volume of the sphere, giving your answer accurate to two decimal places.

From the diagram, we see that the sphere has a radius of 6.3 cm. Recalling the formula , where is the volume and is the radius of the sphere, we can substitute into the right-hand side to get

Therefore, the volume of the sphere rounded to two decimal places is 1โโโ047.39. Since the radius was given in centimetres, this volume is in cubic centimetres; the volume of the sphere is 1โโโ047.39 cm3 to two decimal places.

Notice that because is a nonterminating decimal number, we usually have to round the answer shown on our calculator. For this reason, some questions ask for their answers to be given in terms of . As well as being neater to express, these answers have the advantage of being exact.

### Example 2: Calculating the Volume of a Sphere in terms of Pi

A sphere has a diameter of 18 cm. Work out the volume of the sphere, giving your answer in terms of .

Here, we are given the diameter of the sphere, 18 cm, which is twice its radius. Now, recall the formula , where is the volume of the sphere and is the radius. To apply this formula to work out the volume, we first need to calculate the radius, so we halve the diameter to get . Then, substituting for in the formula, we have

Note that because the radius, 9, is a multiple of the denominator, 3, we can expand as and reorder the expression on the right-hand side to

This then simplifies to

No further calculation is needed because this answer is a multiple of . Since the diameter is given in centimetres, the volume of the sphere must be cm3.

The formula for the volume of a sphere contains only two variables, and . This means that if we are given the volume of a sphere, then we can always work backward to find its radius. Once we have worked out the radius, if needed, we can double this value to obtain the diameter. The next example shows how to rearrange the formula to solve this type of problem.

### Example 3: Finding the Diameter of a Sphere given Its Volume

Find the diameter of a sphere whose volume is 113.04 cm3. (Take ).

First, we recall the formula to calculate the volume, , of a sphere of radius :

We have the volume , together with the approximation 3.14 for . Substituting both of these values into the formula, we get which simplifies to

Multiplying both sides by 75 gives us and dividing through by 314 gives us

To find , we need to take cube roots of both sides: so , which is the same as .

The unit of volume is the cubic centimetre, so the radius is in centimetres. Doubling the value of the radius, we find that the diameter of the sphere is 6 cm.

Notice that in the above example we substituted the value for , the volume of the sphere, into the formula and then rearranged to find the value of the radius, . An alternative approach is to rearrange the formula to make the subject and then substitute for directly. Here, we outline how to derive this formula for the radius in terms of the volume.

Starting with the original formula and rewriting the right-hand side to include multiplication signs, we have

Multiplying both sides by 3, we get and dividing through by gives

Finally, we take cube roots of both sides: so , which is our formula for the radius. If we were to substitute into this formula, together with the approximation 3.14 for , then we would get , the same value for the radius as obtained in the previous example.

### Formula: Radius of a Sphere given Its Volume

The radius, , of a sphere of volume is given by the formula .

When a sphere is cut by a plane that passes through its center, the intersection of the sphere and the plane is called a great circle. Another way to think of this is that a great circle of a sphere is a circle that cuts the sphere exactly in half, forming two hemispheres. This means that the radius of the great circle will also be the radius of the sphere (as well as of each hemisphere).

We can use the information about great circles to calculate the volume of the corresponding spheres or hemispheres. (Note that because the volume of a hemisphere is exactly half that of the associated sphere, we can adjust the expression for the volume of a sphere of radius to for the volume of a hemisphere of radius .)

### Example 4: Calculating the Volume of a Hemisphere

Work out the volume of the hemisphere, giving your answer accurate to two decimal places.

The diagram includes a great circle with radius 8.3 m. Now, recall the formula , where is the volume of a sphere and is its radius. Since the radius of the great circle is also the radius of the corresponding sphere, then substituting into the formula, we have that the volume of this sphere is

To work out the volume of the hemisphere above, we divide this answer by 2, which gives . The radius of the great circle was given in metres, so the hemisphere must have volume 1โโโ197.55 m3 rounded to two decimal places.

It is straightforward to adapt the above method to find other fractions of a sphere. For instance, if asked to find the volume of a third of a sphere of radius , we could take the expression for the volume of a sphere of radius and multiply it by . This tells us that the volume of a third of a sphere of radius is .

Finally, we sometimes meet questions with a real-life context that are given as word problems, without diagrams. In cases like this it is important to read the question carefully and work out exactly what we are being asked to find. Once we have done this, we should be able to apply the formula, perhaps more than once, in order to obtain our answer.

### Example 5: Solving a Word Problem Involving Volumes of Spheres

A sphere of metal with radius 14.1 cm was melted down and formed into 4 equal spheres. Find the radius of one of the smaller spheres, giving your answer to the nearest centimetre.

Our first step is to work out the total volume of metal in the original sphere. Recall that the volume of a sphere of radius is given by the formula . Applying this formula with , we get

Dividing this total volume by 4, we calculate the volume of each smaller sphere to be cm3.

Next, we apply the formula to one of the smaller spheres, using the volume to work out the corresponding radius:

Multiplying both sides by 3, we get and dividing through by gives

Finally, we take the cube roots of both sides to get , which gives an answer of 9 when rounded to the nearest centimetre. Therefore, the length of the radius of one of the smaller spheres is 9 cm.

Let us finish by recapping some key concepts from the explainer.

### Key Points

• The volume, , of a sphere of radius is given by the formula .
• Since the formula contains only two variables, and , if we are given the volume of a sphere, we can always work backward to find its radius (or diameter).
• A great circle of a sphere is a circle that cuts the sphere exactly in half, forming two hemispheres. A great circle and its corresponding sphere have the same center and radius.
• We can use the formula to find the volume of a hemisphere or other fractions of a sphere, including in real-world questions presented as word problems.

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