finding-the-area-and-volume-of-a-hemisphere

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A hemisphere is sphere that has been cut in half. When you cut the sphere in half you are left with the great circle, plus half of a sphere. This fact can be used to find the area, and the volume of a hemisphere. The video works several example problems in which the area and volume of a hemisphere is calculated.

Transcript

The last concept in this unit is hemispheres. This unit on hemispheres is building off of spheres. If you understand spheres hemispheres are really easy.

What is a hemisphere ?

You take a sphere and slice it in half.

You have the top and you have the bottom.

Like taking a orange and sharing with a sibling, you get half and they get half.

So a hemisphere is half a sphere.

To find the area of a hemisphere I need you to think for a second.

If you have half of an orange you have half of the surface area that is the skin of the orange, but don't forget that surface area also includes the flat part.

This is the face of the orange.

When you are finding the surface area of a hemisphere you not only count the curved part, you also have to count the flat part.

What is the flat part on a hemisphere, or an orange?

It is the great circle.

What is the area of the great circle? pi r squared

It is the area of a circle in relation to the sphere.

It is a round curve.

The area of a sphere is 4pi radius squared

However, we want a hemisphere so we will divide this in half.

That will be 2pi r squared.

That formula is the area of the curved part of the hemisphere.

We also have to account for the flat part of the hemisphere and that will be 2pi r squared, and we have to add one more circle to it for the flat part.

We end up with 2 pi r squared + pir squared, which gives us 3 pi r squared.

This is the formula for the surface area of a hemisphere.

including the flat part.

I wanted to show you where that came from.

Now let's look at the volume.

The volume is different.

You have an orange, slice it in half and you get half and they get half.

Is there any extra volume?

No so we will take 4/3 pi r cubed times 1/2.

We are just taking half of a sphere.

This ends up being 4/3 which is simplified to 2/3 pi radius cubed for the volume of a hemisphere.

Again the volume formula of a hemisphere is 2/3 pi radius cubed.

We get this be reducing 4/3

Let's work a forward problem.

Given a radius of 6, find the surface area and volume of a hemisphere.

The surface area will be 3 pi times the radius which is 6 squared.

6 squared is 36

so 3 times pi times 36 is 108 pi inches squared.

That is how you find surface area of a hemisphere given radius.

The volume will be similar you are just plugging in 2/3 formula.

2/3 times pi radius(6) cubed, which is 216

So 2/3 times pi times 216 and that equals 144 pi and this will be inches cubed because we are talking about volume.

Another option for the volume of a hemisphere is to take the volume of a whole sphere and divide it in half.

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Tagged under: hemisphere,sphere,great circle,volume hemisphere,geometry,Area (Dimension),Volume (Dimension),moomoomath,formula surface area hemisphere,formula volume hemisphere

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