8.5VolumeofRoundedObjects
A basic definition of volume is how much space an object takes up. Since this is a three-dimensional
measurement, the unit is usually cubed. For example, we might talk about how many cubic feet of water are in a
pool (or N> ) or how many cubic millimeters of ink are in an ink pen (or ). Let’s explore some previously learned
concepts about volume before diving into cylinders.
Volume Basics
Perhaps the most recognizable formula for volume comes from a rectangular prism. This easily
remembered formula is Ì = œHℎ, or the volume is the length times the width times the height. So in the example
below, we see that the volume is 180 E.
Ì = œHℎ
Ì = 12 ∗ 3 ∗ 5
Ì = 180 E
5 E.
12 E.
3 E.
This is a nice neat formula, but unfortunately it promotes the idea that all we do to find the volume is
multiply all the numbers we see. This simply isn’t true, so we need to know where this formula came from.
Hopefully the idea of length times width sounds familiar. We should recognize that as the area formula for
a rectangle, but which rectangle specifically on our rectangular prism above? It is the bottom face of the prism, the
12 by 3 rectangle. This bottom face is known as the base of the prism. The base shapes of a prism are the shapes
that are congruent and parallel, or the top and bottom in our prism above. This means that we could rewrite the
volume formula to say Ì = jℎ or volume equals the area of the base times the height of the prism.
6 0.
8 0.
20 0.
This formula is much more useful because it will work for all prisms. For
example, consider the triangular prism below. In this case the base shape is a
triangle. Since the height of a prism is the perpendicular distance between the
bases, we know the height is 20 cm. So we apply our formula of Ì = jℎ by
finding the area of the triangle and multiplying by 20, the height.
8∗6
 20 = 2420 = 480 0
Ì = jℎ = Á
2
In essence, what we should take from this is that the volume is like taking the base shape and stacking up
more and more of those shapes until it hits the height of the prism. In other words, Ì = jℎ is a very nice and widely
applicable formula.
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Volume of a Cylinder
So let’s take that concept of the area of the base shape multiplied by the height and transfer it to the
cylinder. While the cylinder is not a prism, it is similar. The base is a circle, which we know the area to be Çy , and
the height is the distance between the circles. This means that we can still use the formula Ì = jℎ or in this case,
Ì = Çy ℎ.
5 .
Ì = Çy ℎ
Ì = Ç(5) (10
Ì = Ç25)(10
10 .
Ì = Ç250)
Ì = 250Ç ≈ 250 ∗ 3.14 ≈ 785 This means that this particular cylinder takes up approximately 785 cubic meters of space. Another way to
think about it is that about 785 little cubes that are a meter on each side would fit inside this cylinder.
In Terms of Pi
Typically it makes sense to plug in the approximation of 3.14 for pi. This gives us an idea of the actual
number or size we’re dealing with. However, sometimes a problem will ask for an answer in terms of pi. That
means to not actually plug in 3.14 for pi. Just leave pi in the answer. So in the above example, the volume would
just be 250Ç . Let’s look at another example.
3 0.
7 0.
Ì = Çy ℎ
Ì = Ç(3) (7)
Ì = Ç(9)(7)
Ì = Ç(63)
Ì = 63Ç 0
So our final answer in terms of pi is 63Ç 0 . Of course we can always plug in 3.14 for pi to get an
approximate answer which in this case is ≈ 197.82 0.
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Volume of a Cone
Cones are similar to prisms and therefore we will use a variant of the Ì = jℎ formula. In fact, the cone is
nearly the same as a cylinder except that a cone only has one base shape. At the top of the cone is a vertex instead
of a second congruent circle. This means that a cone with the same exact circular base and height as a cylinder will
hold less. The question is how much less?
We may recall from previous courses that the volume of a pyramid uses the formula Ì = jℎ. Since a cone
is very similar to a pyramid, it would be reasonable to expect to use the same formula. It turns out this is correct.
The actual proof for this takes some calculus, so we’ll take it on faith for right now. Could you design an experiment
to gather some evidence that this formula will work for a cone?
Let’s find the volume of this cone.
1
Ì = jℎ
3
12 5
1
Ì = Ç(5) (12)
3
1
Ì = Çy ℎ
3
1
Ì = Ç(25)(12)
3
1
Ì = Ç(300
3
Ì = 100Ç ≈ 314 Again we can leave our answer in terms of pi or use 3.14 to approximate the answer.
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Volume of a Sphere
While pyramids and cones share a volume formula, spheres have their own. Spheres use the formula
Ì=
Çy .
Applying this formula is similar to applying the previous volume formulas. In the biz we call this “plug
and chug” math.
Let’s find the volume of this sphere.
4
Ì = Çy 3
Ì = Ç(30
30 N>
4
Ì = Ç27000
3
Ì = 36000Ç ≈ 113040 N> Keep in mind that we are still following order of operations. So we take the radius cubed first. After that
we apply the commutative property and multiply the radius cubed by the fraction value of . The last thing we do
is multiply by pi because sometimes we want to leave it in terms of pi. If we don’t want to leave it in terms of pi,
we multiply by the approximation of pi which is 3.14. Finally, don’t forget the unit for the answer.
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Lesson 8.5
Answer the following questions either using Í U {. r² or giving your answer in terms of Í. Round your answer
to the nearest hundredth where necessary.
1. Find the volume of a cylinder with a radius of 3 E and a height of 10 E.
y
ℎ
2. Find the volume of a cylinder with a radius of 10 and a height of 2 .
3. Find the volume of a cylinder with a radius of 5 0 and a height of 15 0.
4. Find the volume of a cylinder with a diameter of 22 and a height of 5 .
5. Find the volume of a can of green beans with a radius of 3 0 and a height of 8 0.
6. Find the volume of a cylindrical can of oatmeal with a radius of 8 0 and a height of 45 0.
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7. Find the volume of a cone with a radius of 3 E and a height of 10 E.
ℎ
y
8. Find the volume of a cone with a radius of 10 and a height of 3 .
9. Find the volume of a cone with a diameter of 4 N> and a height of 9 N>.
10. Find the volume of a cone with a diameter of 18 E and a height of 9 E.
11. Find the volume of a waffle cone for ice cream with a radius of 4 0 and a height of 12 0.
12. Find the volume of a cone birthday hat with a radius of 2 E and a height of 9 E.
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13. Find the volume of a sphere with a diameter of 6 E.
y
14. Find the volume of a sphere with a diameter of 18 .
15. Find the volume of a sphere with a radius of 6 0.
16. Find the volume of a sphere with a radius of 12 .
17. Find the volume of a mini-basketball with a radius of 3.5 E.
18. Find the volume of a gumball with a radius of 3 .
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