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## Basic geometry and measurement

### Course: Basic geometry and measurement > Unit 10

Lesson 3: Volume of cylinders, spheres, and cones# Volume of a sphere

The formula for the volume of a sphere is V = 4/3 π r³, where V = volume and r = radius. The radius of a sphere is half its diameter. So, to calculate the surface area of a sphere given the diameter of the sphere, you can first calculate the radius, then the volume. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- When Mr. Khan said the volume formula at0:40, is the 4/3 rounded a bit or is it exact?(78 votes)
- Since the Volume of a Sphere is V=(4/3)πr^3, we are using the rational number 4/3 to give us the EXACT solution. Notice however that in this video, after Mr. Khan does his substitution, he uses his calculator to find an APPROXIMATE solution of V=(4/3)π(7)^3=1436.8. Had the problem specified to find the EXACT volume, then we would just substitute, V=(4/3)π(7)^3 and simplify to V=(4/3)π(343)=(1372/3)π, where we do NOT perform the division.(19 votes)

- Where do you get 4/3 from?!(36 votes)
- Great question!! The 4/3 isn't so obvious and requires some work to derive.

Consider the following two figures:

Figure 1: the top half of a sphere with radius r.

Figure 2: a cylinder with radius r and height r, but with a cone (with point on bottom at the center of the cylinder's bottom base) with radius r and height r removed from it.

From the volume formulas for a cylinder and a cone, the volume of Figure 2 is

pi r^2 * r - (1/3) pi r^2 * r = (2/3) pi r^3.

Now we need to compare the areas of the horizontal cross sections of Figure 1 and Figure 2 at any given height h above the bottom. Once we show that these cross sections have equal areas at every height, then Cavalieri's principle would imply that the volumes of Figure 1 and Figure 2 are equal (since the overall heights of the two figures are equal, specifically to r).

In Figure 1, the cross section is a circle with radius sqrt(r^2 - h^2) from the Pythagorean Theorem (hypotenuse is r, one leg is h, and the other leg is the cross section's radius).

So the area of the cross section at height h is pi[sqrt(r^2 - h^2)]^2 = pi(r^2 - h^2).

In Figure 2, the cross section is a ring-shaped region with outer radius equal to r (from the cylinder, since each cross section's radius is the cylinder's radius) and inner radius equal to h (from the cone, since in a cone with equal height and radius, each cross section's radius equals its height above the bottom point).

So the area of the cross section at height h is pi r^2 - pi h^2 = pi(r^2 - h^2).

Therefore, these cross sections have equal areas at every height. So Figure 1 and Figure 2 have the same volume.

Since we have found that the volume of Figure 2 is (2/3) pi r^3, the same is true for Figure 1, which is a hemisphere of radius r.

Therefore, the volume of a full sphere is (4/3) pi r^3.

(By the way, if you take calculus later, you will be able to derive this formula in another way by finding an integral. The volume of a full sphere is integral -r to r of pi(r^2 - x^2) dx. )(82 votes)

- When I was doing my math homework, I was going along with the video and keying in the numbers on my page into the calculator. I ran into one problem. I didn't get the right answer and now I don't know what to do. My radius was 12, but my answer was 7238.229474. It's wrong though. I don't know what to do. Please help!(9 votes)
- V = 4/3 π r^3, so first check and see if the problem asks for the answer in terms of pi which would be 2304 π. If the question asks for the approximate answer, and we multiplied 2304 times π, your answer would be correct, so you should look at where they are asking you to round the number to, your rounding to the nearest millionth will almost always be overkill, the normal questions asking for rounding answers is to nearest whole number, nearest tenth, or nearest hundredth. So since the math is correct, then my assumption is that you did not answer the question to the accuracy that was asked (or in terms of pi).(18 votes)

- Is it easier to use 3.14 while solving, or pi? My math teacher lets us do either, but I'm not sure which one would end up being more accurate.(5 votes)
- Just using the symbol π is infinitely more accurate than writing 3.14. The only catch is that leaving your answer in terms of π doesn't give you a decimal expansion of your answer. Experimental scientists and engineers will often use 3.14 (or even just 3), while mathematicians and more theoretical-focused people will use π.

It just depends on what you want out of your answer.(21 votes)

- How do you find the surface area when you only have the volume?(6 votes)
- I’m assuming you’re asking about finding a sphere’s surface area, given its volume.

Substitute the given volume for V in the equation V = (4/3)pi r^3 and solve for the radius r. Solving for r involves dividing both sides by (4/3)pi and then taking the cube root of both sides.

Once you find r, substitute your value of r into the equation S = 4pi r^2 to find the surface area S.(13 votes)

- How do you deduce the formula? It's really important to me, please help! (I know it includes geometry when you deduce, because I need geometry when I'm deduce the formula of area of circles)(8 votes)
- The formula for volume of a sphere is... v=4/3πr^3

and the formula for surface area of a sphere is... a=4πr^2(8 votes) - What is the formula for finding the volume of a sphere with the same radius and height of a cylinder(vis- versa)? I have been searching the web and still have not found a clear answer. Please help me.

Thank you!(5 votes)- You do need to be more specific, the volume of a sphere is V = 4/3 π r^3, it does not need to be related to a cylinder. So if you know the radius, you can calculate the volume. The volume of a cylinder with the same radius and with a height of 2r (since it would be the diameter across) would be V = π r^2 h = 2π r^3. So the empty space of a sphere placed in a cylinder would be V = 2πr^3 - 4/3πr^3 = 2/3 π r3.(5 votes)

- what is the volume of 1 and a half sphere(4 votes)
- The volume of a sphere is (4/3)pi r^3. So the total volume of a sphere and a hemisphere with the same radius is (3/2)(4/3)pi r^3 = 2pi r^3.(5 votes)

- How do you find the surface area of a sphere?(3 votes)
- the formula is 4π r^2.(6 votes)

## Video transcript

Find the volume of a sphere with
a diameter of 14 centimeters. So if I have a sphere-- so
this isn't just a circle, this is a sphere. You could view it as
a globe of some kind. So I'm going to shade
it a little bit so you can tell that it's
three-dimensional. They're giving us the diameter. So if we go from one
side of the sphere straight through
the center of it. So we're imagining that we
can see through the sphere. And we go straight
through the centimeter, that distance right over
there is 14 centimeters. Now, to find the volume of a
sphere-- and we've proved this, or you will see a proof for this
later when you learn calculus. But the formula for
the volume of a sphere is volume is equal
to 4/3 pi r cubed, where r is the
radius of the sphere. So they've given
us the diameter. And just like for circles,
the radius of the sphere is half of the diameter. So in this example, our radius
is going to be 7 centimeters. And in fact, the sphere
itself is the set of all points in three
dimensions that is exactly the radius away from the center. But with that out of
the way, let's just apply this radius being 7
centimeters to this formula right over here. So we're going to
have a volume is equal to 4/3 pi times 7
centimeters to the third power. So I'll do that in
that pink color. So times 7 centimeters
to the third power. And since it
already involves pi, and you could
approximate pi with 3.14. Some people even
approximate it with 22/7. But we'll actually just
get the calculator out to get the exact
value for this volume. So this is going to
be-- so my volume is going to be 4 divided by 3. And then I don't want
to just put a pi there, because that might interpret
it as 4 divided by 3 pi. So 4 divided by 3 times pi,
times 7 to the third power. In order of operations,
it'll do the exponent before it does the
multiplication, so this should work out. And the units are going to be
in centimeters cubed or cubic centimeters. So we get 1,436. They don't tell us
what to round it to. So I'll just round it to
the nearest 10th-- 1,436.8. So this is equal to
1,436.8 centimeters cubed. And we're done.