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Surface Area and Volume

Surface Area and Volume

Volume of Cones

The same formula we used for pyramids (V = ⅓Bh) applies to cones, only the base isn't a polygon anymore. It's a circle. Formula-wise, that means we replace B with πr2. That's it.

Sample Problem

Along with a diet and strict workout regimen, your Aunt Bertha decides that any time she eats ice cream, she'll pack it into the cone, but won't have anything above it. The height of the cone is 10 centimeters and the radius is 3 centimeters. If 100 cubic centimeters of ice cream is 200 Calories and the cone itself is 50 Calories, how many Calories are in one of Aunt Bertha's ice cream cones?

First, we can calculate the volume of the cone using our formula.

V ≈ 94.2 cm3

That's how much ice cream Aunt Bertha packs into the cone. To calculate how many Calories she eats per cone, we need to convert the volume to Calories. Don't forget to add the 50 Calories for the waffle cone itself.

Calories = 50 Calories + 188.4 Calories
Calories = 238.4 Calories

Aunt Bertha eats 238.4 Calories per ice cream cone. She can treat herself to one every now and then. She'll work it off on the elliptical at the gym, anyway.

Bonnie Cavalieri worked to bring justice to the oblique cylinders and prisms of the world, but he succeeded in doing so much more. Oblique cones and pyramids also share in the glory of his principle, and today Cavalieri is known as the Knight in Slanting Armor, a hero for all oblique solids.

Except not really. His principle does apply to cones and pyramids just the same, though.

Sample Problem

What is the volume of this oblique cone?

Calculating the volume shouldn't be a problem. We just have to do a little dance, make a little love, and trig it up tonight.

First thing we have to do is find the height of the cone. We know the hypotenuse and the angle opposite the height. We don't need any other sines to tell us what trig function to use.

h = (19 in) × sin(56°)

People might think dogs are man's best friend, but they're wrong. Calculators are.

h ≈ 15.75 in

Now we can use the same formula to find the volume of the cone.

V ≈ 808.2 in3

Danger, Will Robinson! Make sure you remember the difference between surface area and volume. While we can apply Cavalieri's principle to oblique cylinders and pyramids for volume, we can't do the same with surface area.

That's all folks!

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