## Lesson 10: Solids. Surface Areas. An Extra Question with a Prize (IV)

May 23, 2012

### Are you able to prove it?

Think of the net of a cone.

If the base of the cone has radius r and the cone has generatrix g , I told you the lateral area (the area of the circle sector) was πrg

Could you prove it?

There is an extra 0.5 in this lesson if you give the right proof. Anyway your interest and effort will be prized in the global mark of this term.

1. We are going to prove two ways to work out the lateral area of a cone.
First of all, we have to solve the cord:
L = 2πGα/(360°) = 2πr
α=( 2πr•360)/2πG = (r•360)/G
Now we know the degrees but it´s another thing, the important one is:

A = (πG^(2 ) α)/(360°) = πrG (with the process above we know α)
A =πG^2 • (r360°)/G / 360°=
= (πGr•360)/(360°) = πrG
Doing all this processes we know that both formulas we have the same result to work out the lateral area of the cone.

• Good effort Claudia. Anyway, I believe ita can be easier if you set up a rule of three relating areas and the corresponding lenghts of the arcs that limit them, this is:

Let xbe tha area of the circle sector that is the lateral surface of the cone)

πg^2……………….2πg ———–(the circle has radius g)
x ………………..2πr————(the arc has lenght 2πr as the lenght of the bas of the cone).

Solving for x: x=(πg^2 · 2πr)/2πg = πrg

Mónica

2. This is the metod that i am going to prove:
AL=ng^—- complete area of the circle of radius g, times 5/2ng—proportion between the angle spanned by the arc S and the total arc that is the complete circle of the radius g.

AL= ng^ times 5/2ng= ng^ . 2nr/2ng= ngr because we cross out the square of the ng and the two and the n.

n=number pi
g=generatix
^= the square

3. the method is correct, number pi times the radius times the generatrix of the cone= the lateral area of the cone, I´ll prove with the an example:
If I have a cone (right) as radius of the base 5cm, and a generatrix of 20cm, the thing that i have to do is
formula of the lateral area of a cone = n•r•g, so if i aply this formula to my example I´ll get 3.14•5•20 that has as result 314cm2(square centimiters)