## 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

**I told you the lateral area (the area of the circle sector) was**

*g ,*

*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.*

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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)

by Claudia Urbina Viguera May 27, 2012 at 7:31 pmA =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

by mongarcia May 28, 2012 at 2:57 pmThis 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

by Lydia Ascacíbar Rodríguez May 29, 2012 at 4:44 pmg=generatix

^= the square

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:

by Raúl Ortega June 2, 2012 at 7:42 amIf 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)