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.
Example 1
A right cone has a diameter of 6 m and a height of 12 m. What is its volume? |
Example 2
A cone has been removed from the inside of a square pyramid. If the diameter of the cone's base has the same length as the pyramid's base, what's the volume of the resulting solid? |
Exercise 1
Find the volume of the oblique cone.
Exercise 2
Find the volume of the solid.
Exercise 3
The solid is a section of a cone. Find its volume.
Exercise 4
Your little sister wants to dress up as a fairy for Halloween and she needs you to go shopping with her. At the store, she finds this fairy hat in the shape of a cone and wants you to pay for it. You can't help imagining filling the hat with ice-cold water and dumping it on her. If the circumference of the hat's base is 18 inches, how much water would you dump on her?
Exercise 5
The world's largest double-sided pencil is perfectly cylindrical with the tips taking the shapes of identical cones on both ends. If wood weighs 50 pounds per cubic foot, how heavy is the pencil? Assume the pencil is made entirely of wood.
