Nets, volume, and surface area for circular cones
Finding nets, volume, and surface area of circular cones
In this lesson we’ll look at the nets, volume, and surface area of cones.
Cones
A cone has one circular base.
The net of a cone is a full circle for the base, and the a part of another circle for the wall of the cone.
Parts of a cone
You’ll need to know the parts of a cone.
The radius ???r???, height ???h???, and slant height ???l??? of a cone are related by the Pythagorean Theorem.
???r^2+h^2=l^2???
Volume of a cone
The volume of a cone is given by
???V=\frac{1}{3}\pi {{r}^{2}}h???
where ???r??? is the radius and ???h??? is the height of the cone.
Surface area of a cone
The surface area of a cone is given by
???S=\pi rl+\pi {{r}^{2}}???
where ???r??? is the radius and ???l??? is the slant height of the cone.
???L??? is called the lateral area, so we can also write the surface area of the cone as
???S=L+\pi {{r}^{2}}???
How to calculate the surface area of a circular cone
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Finding surface area from the net
Example
When the net of the cone is folded up, what is its surface area?
The formula for the surface area of a cone is
???S=\pi rl+\pi {{r}^{2}}???
In this case, the slant height is ???l=8??? and the radius is ???r=6???. Plugging these into the formula, we get
???S=\pi rl+\pi {{r}^{2}}???
???S=\pi (6)(8)+\pi {{(6)}^{2}}???
???S=48\pi +36\pi???
???S=84\pi \text{ units}^2???
Let’s do a few more examples.
Example
What is the volume of the cone?
The formula for volume is
???V=\frac{1}{3}\pi {{r}^{2}}h???
We already know the radius of the cone is ???3\text{ cm}???, and we need to find the height of the cone in order to find the volume. Remember, the radius, height, and slant height are related by the Pythagorean Theorem.
???r^2+h^2=l^2???
Plugging in, we get
???{{3}^{2}}+{{h}^{2}}={{5}^{2}}???
???9+{{h}^{2}}=25???
???{{h}^{2}}=16???
???h=4???
Now we can use the volume formula.
???V=\frac{1}{3}\pi {{r}^{2}}h???
???V=\frac{1}{3}\pi {{(3)}^{2}}(4)???
???V=\frac{1}{3}(9)(4)\pi???
???V=12\pi \text{ cm}^{3}???
Let’s do one more.
Example
What is the surface area of a cone with a slant height of ???16.8\text{ cm}??? and a diameter of ???16\text{ cm}???? Use ???\pi =3.14???.
The formula for the surface area is
???S=\pi rl+\pi {{r}^{2}}???
The slant height is ???l=16.8???. We can use the diameter to find the radius.
???r=\frac{d}{2}???
???r=\frac{16}{2}=8???
The radius is ???r=8???. Plugging these values into the formula, we get
???S=\pi rl+\pi r^2???
???S=3.14(8)(16.8)+3.14{{(8)}^{2}}???
???S=622.976\text{ cm}^2???