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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn where in the real world you will see cones. Also, you will learn the formulas for finding the surface area and volume of these cones.

When I hear the word 'cones,' I think of ice cream cones, especially the waffle kind. The general shape of these waffle cones for ice cream is that of the cones that we are talking about in this lesson. Mathematically defined, a **cone** is a three-dimensional object with a flat base and one curved side that slants to a tip at the top. Another real-world example of a cone is those orange traffic cones. Can you see how both ice cream cones and the traffic cones have one curved side that slants to a point?

When working with cones, there are three measurements that we need to be concerned about. They are the radius, the height of the cone, and the length of the side from tip to base. We can label these below on a drawing of a cone with *r* for the radius, the distance from the center of the circular base to the edge of the base; *h* for the height of the cone, how tall the cone stands; and *s* for length of the side from tip to base. Our formulas for surface area and volume use these measurements.

Our formula for surface area has two parts, one for the base and one for the curved side. The surface area is the total area of just the surface of our object. The formula for the area of the base is:

*Surface Area Base* = Ï€*r*2

The formula for the curved side is:

*Surface Area Curved Side* = Ï€*rs*

The *s* can be rewritten using the height and radius as:

*s* = âˆš(*r*2 + *h*2)

We need to add both the base and curved side together to get the total surface area, so our complete surface area formula for a cone is:

*Surface Area* = Ï€*r*2 + Ï€*rs*

If our problem gives us the radius and the *s* measurement, the length of the side from tip to base, then we can plug that number directly into the formula for *s*. But if the problem only gives us the height and the radius of the cone, then we would have to rewrite our problem like this:

Surface Area = Ï€*r*2 + Ï€*r* * âˆš(*r*2 + *h*2)

We use this version of the formula if we are only given the height and radius of the cone.

Let's see how this works with a sample problem. We have a cone, and we are told that it has a radius of 3 inches and a height of 10 inches. We see that we are given the *r* and the *h*, but not the *s*. So, this tells me that I need to use the second version of the surface area formula that uses the height instead of the side length. So, I plug in my numbers into this formula; I plug in 3 for *r* and 10 for *h* wherever I see them:

Surface Area = 3.14 * 32 + 3.14 * 3 * âˆš(32 + 102)

Now I go ahead and evaluate to find my answer. I square the 3 and 10 and then I add them together to get 109. I take the square root of this to get 10.44, which I then multiply with the 3 and the 3.14 to get 98.35. I then multiply the other part of 3.14 with the square of 3, which gives me 28.26. I then add the 28.26 and the 98.35 together to get my answer of 126.61 inches squared.

Surface Area = 3.14 * 32 + 3.14 * 3 * âˆš(9 + 100)

Surface Area = 3.14 * 32 + 3.14 * 3 * âˆš(109)

Surface Area = 3.14 * 32 + 3.14 * 3 * 10.44

Surface Area = 3.14 * 32 + 98.35

Surface Area = 3.14 * 9 + 98.35

Surface Area = 28.26 + 98.35

Surface Area = 126.61 inches squared

I remember that area is always squared, so I make sure my answer will always end with my measuring units squared.

Now, if my problem gave me the side length instead, I would go through the same process except I would be using the version of the formula with the *s* instead of the *h*.

For volume, the amount of space inside our object, there is only one version of the formula, which uses the height instead of the side length. The formula for the volume of a cone is:

Volume =

Again, once you have your radius and height, all you need to do is plug these numbers into the formula and evaluate to get your answer.

Let's find the volume of the sample cone that we found the surface area for. Let's see. Our radius for the cone is 3 inches, and our height is 10 inches. We plug these numbers into our formula to get the following.

Volume =

We then evaluate this by squaring the 3 to get 9 and then multiplying that by 3.14 and 10 to get 282.6, which we divide by 3 to get our answer of 94.2 inches cubed.

I recall that volume is always cubed, so I make sure my answer includes my measuring units cubed.

What have we learned? We learned that **cones** are three-dimensional objects with a flat base and one curved side that slants to a tip at the top. In the real world, we see cones such as ice cream waffle cones and traffic cones. The measurements that we are concerned about for cones are the radius, the height, and the length of the side from tip to base.

The formula for finding the surface area of a cone is:

*Surface Area* = Ï€*r*2 + Ï€*rs*

where the *s* can be replaced by: âˆš(*r*2 + *h*2)

if the problem gives you the height instead of the side length.

The formula for volume is:

Volume =

where all you need are the height and radius.

With both of these formulas, all you need is to plug in your appropriate numbers and evaluate to find your answers. Just remember that area is always squared and volume is always cubed.

Finishing this lesson could enable you to:

- Illustrate the shape of a cone and relate it to real-world objects
- Recognize the formulas for surface area and volume of a cone
- Write out the two formulas used to find the surface area

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Geometry: High School15 chapters | 160 lessons

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