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 ๐œ‹(=3.14159โ€ฆ). 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 227. In view of this, it is perhaps not surprising that we meet ๐œ‹ again when we move on to the three-dimensional problem of calculating the volume of a sphere.

Formula: Volume of a Sphere

The volume, ๐‘‰, of a sphere of radius ๐‘Ÿ is given by the formula ๐‘‰=43๐œ‹๐‘Ÿ๏Šฉ.

Since 43 and ๐œ‹ 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.

Answer

From the diagram, we see that the sphere has a radius of 6.3 cm. Recalling the formula ๐‘‰=43๐œ‹๐‘Ÿ๏Šฉ, where ๐‘‰ is the volume and ๐‘Ÿ is the radius of the sphere, we can substitute ๐‘Ÿ=6.3 into the right-hand side to get ๐‘‰=43ร—๐œ‹ร—(6.3)=1047.394โ€ฆ.๏Šฉ

Therefore, the volume of the sphere rounded to two decimal places is 1โ€Žโ€‰โ€Ž047.39. Since the radius was given in centimetres, this volume is in cubic centimetres; the volume of the sphere is 1โ€Žโ€‰โ€Ž047.39 cm3 to two decimal places.

Notice that because ๐œ‹ 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 ๐œ‹. 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 ๐œ‹.

Answer

Here, we are given the diameter of the sphere, 18 cm, which is twice its radius. Now, recall the formula ๐‘‰=43๐œ‹๐‘Ÿ๏Šฉ, where ๐‘‰ is the volume of the sphere and ๐‘Ÿ 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 ๐‘Ÿ=18รท2=9. Then, substituting for ๐‘Ÿ in the formula, we have ๐‘‰=43ร—๐œ‹ร—9.๏Šฉ

Note that because the radius, 9, is a multiple of the denominator, 3, we can expand 9๏Šฉ as 9ร—9๏Šจ and reorder the expression on the right-hand side to 43ร—9ร—๐œ‹ร—9.๏Šจ

This then simplifies to 4ร—3ร—๐œ‹ร—9=4ร—3ร—๐œ‹ร—81=4ร—3ร—81ร—๐œ‹=972๐œ‹.๏Šจ

No further calculation is needed because this answer is a multiple of ๐œ‹. Since the diameter is given in centimetres, the volume of the sphere must be 972๐œ‹ cm3.

The formula for the volume of a sphere contains only two variables, ๐‘‰ and ๐‘Ÿ. 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 ๐œ‹=3.14).

Answer

First, we recall the formula to calculate the volume, ๐‘‰, of a sphere of radius ๐‘Ÿ: ๐‘‰=43๐œ‹๐‘Ÿ.๏Šฉ

We have the volume ๐‘‰=113.04, together with the approximation 3.14 for ๐œ‹. Substituting both of these values into the formula, we get 113.04=43ร—3.14ร—๐‘Ÿ,๏Šฉ which simplifies to 113.04=31475ร—๐‘Ÿ.๏Šฉ

Multiplying both sides by 75 gives us 8478=314ร—๐‘Ÿ,๏Šฉ and dividing through by 314 gives us 27=๐‘Ÿ.๏Šฉ

To find ๐‘Ÿ, we need to take cube roots of both sides: ๏Žข๏Žขโˆš27=โˆš๐‘Ÿ,๏Šฉ so 3=๐‘Ÿ, which is the same as ๐‘Ÿ=3.

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 ๐‘‰, the volume of the sphere, into the formula ๐‘‰=43๐œ‹๐‘Ÿ๏Šฉ and then rearranged to find the value of the radius, ๐‘Ÿ. An alternative approach is to rearrange the formula to make ๐‘Ÿ the subject and then substitute for ๐‘‰ directly. Here, we outline how to derive this formula for the radius in terms of the volume.

Starting with the original formula ๐‘‰=43๐œ‹๐‘Ÿ๏Šฉ and rewriting the right-hand side to include multiplication signs, we have ๐‘‰=43ร—๐œ‹ร—๐‘Ÿ.๏Šฉ

Multiplying both sides by 3, we get 3๐‘‰=4ร—๐œ‹ร—๐‘Ÿ,๏Šฉ and dividing through by 4๐œ‹ gives 3๐‘‰4๐œ‹=๐‘Ÿ.๏Šฉ

Finally, we take cube roots of both sides: ๏Žข๏Žข๏„ž3๐‘‰4๐œ‹=โˆš๐‘Ÿ,๏Šฉ so ๐‘Ÿ=๏„ž3๐‘‰4๐œ‹๏Žข, which is our formula for the radius. If we were to substitute ๐‘‰=113.04 into this formula, together with the approximation 3.14 for ๐œ‹, then we would get ๐‘Ÿ=3, the same value for the radius as obtained in the previous example.

Formula: Radius of a Sphere given Its Volume

The radius, ๐‘Ÿ, of a sphere of volume ๐‘‰ is given by the formula ๐‘Ÿ=๏„ž3๐‘‰4๐œ‹๏Žข.

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 43๐œ‹๐‘Ÿ๏Šฉ for the volume of a sphere of radius ๐‘Ÿ to 23๐œ‹๐‘Ÿ๏Šฉ for the volume of a hemisphere of radius ๐‘Ÿ.)

Example 4: Calculating the Volume of a Hemisphere

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

Answer

The diagram includes a great circle with radius 8.3 m. Now, recall the formula ๐‘‰=43๐œ‹๐‘Ÿ๏Šฉ, where ๐‘‰ is the volume of a sphere and ๐‘Ÿ is its radius. Since the radius of the great circle is also the radius of the corresponding sphere, then substituting ๐‘Ÿ=8.3 into the formula, we have that the volume of this sphere is ๐‘‰=43ร—๐œ‹ร—(8.3)=2395.095โ€ฆ.๏Šฉ

To work out the volume of the hemisphere above, we divide this answer by 2, which gives 1197.547โ€ฆ. The radius of the great circle was given in metres, so the hemisphere must have volume 1โ€Žโ€‰โ€Ž197.55 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 ๐‘Ÿ, we could take the expression 43๐œ‹๐‘Ÿ๏Šฉ for the volume of a sphere of radius ๐‘Ÿ and multiply it by 13. This tells us that the volume of a third of a sphere of radius ๐‘Ÿ is 13ร—43๐œ‹๐‘Ÿ=49๐œ‹๐‘Ÿ๏Šฉ๏Šฉ.

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.

Answer

Our first step is to work out the total volume of metal in the original sphere. Recall that the volume ๐‘‰ of a sphere of radius ๐‘Ÿ is given by the formula ๐‘‰=43๐œ‹๐‘Ÿ๏Šฉ. Applying this formula with ๐‘Ÿ=14.1, we get 43ร—๐œ‹ร—(14.1)=11742.104โ€ฆ.๏Šฉ

Dividing this total volume by 4, we calculate the volume of each smaller sphere to be 2935.526โ€ฆ cm3.

Next, we apply the formula to one of the smaller spheres, using the volume to work out the corresponding radius: 2935.526โ€ฆ=43ร—๐œ‹ร—๐‘Ÿ.๏Šฉ

Multiplying both sides by 3, we get 8806.578โ€ฆ=4ร—๐œ‹ร—๐‘Ÿ,๏Šฉ and dividing through by 4๐œ‹ gives 700.805โ€ฆ=๐‘Ÿ.๏Šฉ

Finally, we take the cube roots of both sides to get ๐‘Ÿ=8.882โ€ฆ, 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, ๐‘‰, of a sphere of radius ๐‘Ÿ is given by the formula ๐‘‰=43๐œ‹๐‘Ÿ๏Šฉ.
  • Since the formula contains only two variables, ๐‘‰ and ๐‘Ÿ, 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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