# I cannot solve this problem about surface area of a cone

this is the question that i had problems with

A cone with a radius of 5cm has a surface area of 2000$\pi$ cm${}^2$. what is the perpendicular height of this cone?

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– 1015 Mar 3 '13 at 2:18

## 3 Answers

Let surface area of cone be $S$.

$$S = \pi \cdot r \cdot l$$ where, $r$ = radius of base and $l$ = slant height.

Again, we have $l = \sqrt{r^2 + h^2}$, where $h$ will be the perpendicular height of cone. Thus,

$$l = \dfrac{S}{\pi r} = \dfrac{2000 \pi}{5\pi} \text{ cm} = 400 \text{ cm}$$

And, $$h = \sqrt{400^2 - 5^2} = 25 \sqrt{255} = 399.218 \text{ cm}$$

Alternatively, the area can also mean the base circular part of cone(normally I think of the cone as a conical hats, and hence, don't consider the bottom area).

In this case, we'll have(using same notations as above):

\begin{align} S &= \pi \cdot r \cdot ( r + l ) \\ 2000 \pi &= 5 \pi ( 5 + l ) \\ l &= \dfrac{2000}{5} - 5 \\ l &= 395 \text{ cm} \end{align}

And, we'll have the height as:

$$h = \sqrt{395^2 - 5^2} = 20 \sqrt{390} = 394.968 \text{ cm}$$

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I think in most cases the surface area of a cone is intended to include the base, whereas the term "lateral area" is without the base. – Zev Chonoles Mar 3 '13 at 2:50
@ZevChonoles Updated reply with the reason, why the base wasn't included. – hjpotter92 Mar 3 '13 at 3:06

As user julien commented above, the formula for the surface area $S$ of a cone with radius $r\text{ cm}$ and lateral height (a.k.a. slant height) $l\text{ cm}$ is $$S=\pi r^2+\pi rl.$$ You know the value of $S$ and $r$, so we can solve for $l$: $$2000\pi\text{ cm}^2=25\pi\text{ cm}^2+5l\pi\text{ cm}^2 \implies l=\frac{2000\pi-25\pi}{5\pi}=395$$ However, you want to find the perpendicular height of the cone. Let's say it is $h\text{ cm}$. Then looking at a vertical cross-section of the cone, we'd see a right triangle with one leg whose length is $5\text{ cm}$, whose other leg is $h\text{ cm}$, and whose hypotenuse is $395\text{ cm}$. Now use the Pythagorean theorem to solve for $h$.

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The formula for the surface area, S, of a cone is:

$$S = πr^2 + πrl$$

With r being the radius and l being the length of the cone. We are given the surface area and radius so we can plug those values into our formula:

$$2000pi = (5^2)pi + 5pi*l$$

Solve for l to get:

$$(2000pi-25pi)/(5pi) = l$$

$$395 = l$$

If you look at the shape of a cone you will notice that the perpendicular (i.e right angled) height forms a right triangle with the radius of the cone being the base and the length of the cone being the hypotenuse. Using the Pythagorean theorem and plugging in our known values we will be able to solve for h, the height of the cone.

$$r^2 + h^2 = l^2$$

$$5^2 + h^2 = 395^2$$

$$h^2 = 156025 - 25$$

$$h^2 = 156000$$

$$h = 394.97$$

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Everything is fine, except your last step is wrong: we have $156000<1000000$, so $$\sqrt{156000}<\sqrt{1000000}=1000$$ so there's no way $\sqrt{156000}=1249$. In fact, using a calculator, I get that $$\sqrt{156000}\approx 394.968$$ – Zev Chonoles Mar 3 '13 at 3:02
Also, you can find some good starting points on how to format mathematics on the site here. This AMS reference is very useful. If you need to format more advanced things, there are many excellent references on LaTeX on the internet, including StackExchange's own TeX.SE site. – Zev Chonoles Mar 3 '13 at 3:04
Thanks and thanks! – Johanna Mar 3 '13 at 3:10