5.4.6: Finding Cone Dimensions
- Page ID
- 36039
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Lesson
Let's figure out the dimensions of cones.
Exercise \(\PageIndex{1}\): Number Talk: Thirds
For each equation, decide what value, if any, would make it true.
\(27=\frac{1}{3}h\)
\(27=\frac{1}{3}r^{2}\)
\(12\pi =\frac{1}{3}\pi a\)
\(12\pi =\frac{1}{3}\pi b^{2}\)
Exercise \(\PageIndex{2}\): An Unknown Radius
The volume \(V\) of a cone with radius \(r\) is given by the formula \(V=\frac{1}{3}\pi r^{2}h\).
The volume of this cone with height 3 units and radius \(r\) is \(V=64\pi\) cubic units. This statement is true:
\(64\pi =\frac{1}{3}\pi r^{2}\cdot 3\)
What does the radius of this cone have to be? Explain how you know.
Exercise \(\PageIndex{3}\): Cones with Unknown Dimensions
Each row of the table has some information about a particular cone. Complete the table with the missing dimensions.
| diameter (units) | radius (units) | area of the base (square units) | height (units) | volume of cone (cubic units) |
|---|---|---|---|---|
| \(4\) | \(3\) | |||
| \(\frac{1}{3}\) | \(6\) | |||
| \(36\pi\) | \(\frac{1}{4}\) | |||
| \(20\) | \(200\pi\) | |||
| \(12\) | \(64\pi \) | |||
| \(3\) | \(3.14\) |
Are you ready for more?
A frustum is the result of taking a cone and slicing off a smaller cone using a cut parallel to the base.
Find a formula for the volume of a frustum, including deciding which quantities you are going to include in your formula.
Exercise \(\PageIndex{4}\): Popcorn Deals
A movie theater offers two containers:
Which container is the better value? Use 3.14 as an approximation for \(\pi\).
Summary
As we saw with cylinders, the volume \(V\) of a cone depends on the radius \(r\) of the base and the height \(h\):
\(V=\frac{1}{3}\pi r^{2}h\)
If we know the radius and height, we can find the volume. If we know the volume and one of the dimensions (either radius or height), we can find the other dimension.
For example, imagine a cone with a volume of \(64\pi\) cm3, a height of 3 cm, and an unknown radius \(r\). From the volume formula, we know that
\(64\pi =\frac{1}{3}\pi r^{2}\cdot 3\)
Looking at the structure of the equation, we can see that \(r^{2}=64\), so the radius must be 8 cm.
Now imagine a different cone with a volume of \(18\pi \) cm3, a radius of 3 cm, and an unknown height \(h\). Using the formula for the volume of the cone, we know that
\(18\pi =\frac{1}{3}\pi 3^{2}h\)
so the height must be 6 cm. Can you see why?
Glossary Entries
Definition: Cone
A cone is a three-dimensional figure like a pyramid, but the base is a circle.
Definition: Cylinder
A cylinder is a three-dimensional figure like a prism, but with bases that are circles.
Definition: Sphere
A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.
Practice
Exercise \(\PageIndex{5}\)
The volume of this cylinder is \(175\pi\) cubic units.
What is the volume of a cone that has the same base area and the same height?
(From Unit 5.4.5)
Exercise \(\PageIndex{6}\)
A cone has volume \(12\pi\) cubic inches. Its height is 4 inches. What is its radius?
Exercise \(\PageIndex{7}\)
A cone has volume \(3\pi\).
- If the cone’s radius is 1, what is its height?
- If the cone’s radius is 2, what is its height?
- If the cone’s radius is 5, what is its height?
- If the cone’s radius is \(\frac{1}{2}\), what is its height?
- If the cone's radius in \(r\), then what is the height?
Exercise \(\PageIndex{8}\)
Three people are playing near the water. Person A stands on the dock. Person B starts at the top of a pole and ziplines into the water, then climbs out of the water. Person C climbs out of the water and up the zipline pole. Match the people to the graphs where the horizontal axis represents time in seconds and the vertical axis represents height above the water level in feet.
(From Unit 5.2.4)
Exercise \(\PageIndex{9}\)
A room is 15 feet tall. An architect wants to include a window that is 6 feet tall. The distance between the floor and the bottom of the window is \(b\) feet. The distance between the ceiling and the top of the window is \(a\) feet. This relationship can be described by the equation \(a=15-(b+6)\)
- Which variable is independent based on the equation given?
- If the architect wants \(b\) to be 3, what does this mean? What value of \(a\) would work with the given value for \(b\)?
- The customer wants the window to have 5 feet of space above it. Is the customer describing \(a\) or \(b\)? What is the value of the other variable?
(From Unit 5.2.1)