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Lesson 7: Volume

# 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.

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• When Mr. Khan said the volume formula at , is the 4/3 rounded a bit or is it exact?
• 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.
• Where do you get 4/3 from?!
• 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. )
• 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!
• 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.
• 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.

• How do you find the surface area when you only have the volume?
• 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.
• 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)
• 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!
• 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.
• 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