Video: Finding the Volume of a Sphere given the Diameter

What is the volume of a sphere whose diameter is 30?

02:48

Video Transcript

What is the volume of a sphere whose diameter is 30?

We begin by recalling the key formula we need for calculating the volume of a sphere. It’s four-thirds 𝜋𝑟 cubed, where 𝑟 is the radius of the sphere. In this question, we haven’t been given the radius of the sphere. Instead, we’ve been told that its diameter is 30 units. That’s no great problem though because we know the relationship between the radius and diameter of a sphere. The diameter is twice the radius, or equivalently we can say that the radius is half the diameter.

So to find the radius of this sphere, we simply need to halve the measurement we were given for the diameter. The radius is 30 over two, which is 15 units.

Now that we know the radius, we can substitute directly into our formula for the volume of the sphere, giving four-thirds 𝜋 multiplied by 15 cubed. Now let’s consider how we could simplify this answer if we didn’t have access to a calculator.

We can begin by writing 15 cubed as 15 times 15 times 15. And then we know that three is a factor of 15. So we can cancel the three in the denominator with one factor of three in one of the 15s in the numerator, giving four 𝜋 multiplied by five multiplied by 15 multiplied by 15. We know that we can multiply in any order. So reordering the value slightly, this is equivalent to four multiplied by five multiplied by 𝜋 multiplied by 15 multiplied by 15.

Now four multiplied by five we know is equal to 20. And 15 multiplied by 15 we should be able to work out or we should know. It’s one of our square numbers. 15 squared is equal to 225. So our calculation becomes 20 multiplied by 𝜋 multiplied by 225. We can then think of 20 as 10 multiplied by two. So we have 10 multiplied by two multiplied by 𝜋 multiplied by 225. Multiplying two by 225 gives 450, and then multiplying by 10 gives 4,500. So we can simplify our answer to 4,500𝜋.

Now notice that we didn’t use a calculator at any point in this question. And this is called giving our answer as a multiple of 𝜋. Doing so means that we can answer questions about spheres and circles when we don’t have access to a calculator.

We weren’t given any units for the diameter in this question. So the units for our volume will just be general cubic units. We’ve found then that the volume of a sphere whose diameter is 30 is 4,500𝜋 cubic units.

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