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Surface Area and Volume

Surface Area and Volume

Volume of Spheres

Everyone knows the moon isn't really made of cheese. But what if it were? Picture it. Sure, we'd have a bit of an issue with the ocean tides, but think of all that cheese. We could end world hunger (assuming no one was lactose-intolerant or vegan).

How much cheese would it even be? Well, we can find out using the volume of a sphere.

To find the volume of a sphere, we can chop it up into an infinite number of pyramids. It'll take a while (more like forever), so let's not and say we did. But if we're going to say we did, we better make it convincing.

If we make the base of our pyramids the surface of the sphere and the vertex the very center, our radius ends up being the height of the pyramid. If we added up all these pyramids, we'd end up with an equation that looks like this:

Since our heights all equal the radius, we can replace h with r and factor it out.

Now we've got a hive of B's. All the bases added together equal the surface of the sphere, and we already have the formula for the surface area.

Simplifying it slightly, we have our volume formula.

We have our formula, so let's get down to business...to defeat the Huns.

Sample Problem

What is the volume of this sphere if its radius is 4 inches long?

Its radius is 4 inches and luckily, that's all we need to know.

V ≈ 268.1 in3

Piece of cake, right? Don't worry. We're hip and with-it. We can switch it up, yo.

Sample Problem

What is the diameter of a sphere whose volume is 11,494 cubic feet?

Since we're looking for the radius, it's probably best if we isolate for that first.

Now we can plug in the volume and solve for that missing radius.

Make sure we cube root that mess and not square root.

r ≈ 14 ft

We can find volume using the radius and vice versa. As bountiful (or bouncy-ful) as balls might be, there are other solids we should take into account.

Hemispheres, for instance. Since a hemisphere is exactly half of a sphere, its volume should be exactly half the volume of a sphere. That makes exactly 100% sense.

Sample Problem

What's the volume of this hemisphere?

We've got half a sphere on our hands, so it stands to reason that we wanna cut our volume formula in half.

Good deal. Now let's plug in that radius and get cracking.

V ≈ 1526.8 cm3

Our triumph speaks volumes.

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