how is the formula for finding the surface area of a cone is derived?

Question

that is how the formula "pi *r*l" is derived

Answer 1

How much math have you had?

It can be done easily with calculus.

Imagine a wedding cake as an aproximation for a cone. One could easily measure the surface area of the sides of the wedding cake as each layer is just a cylinder, thus the area is 2πrh, where h is the height of the layer. If you add up the area of all the layers you get an approximation of the area of the cone. Now, as you make your imaginary wedding cake have more and more layers the approximation gets better until eventually you have an infinate number of layers each of height zero. Calculus lets you add all those layers.

The actual calculs follows but you can think about it like this. The bottom layer is 2πrh where r is the radius of the base of the cone and the top layer is 2πrh where r is zero (at the very tip). If you average the two radii (r and 0) you get r/2, thus A = 2π(r/2)l = πrl.

Answer 2

You can think of it as composed of a huge number of very thin slices, each of which can be approximated as a triangle so that the surface is 1/2 l h, where h is the length of the short edge. Those short edges add up to the base of the cone which is 2 pi r, so the total surface is

(1/2 l) * (2 pi r) = pi r l

Answer 3

Formula to find the Area of the Circle = pi. r ^ 2

(where pi = 22/7, r is the radius of the circle

Formula to find the area of a segment of the circle = Pi. r* l

Pi = 22/7; r = radius of the base (circle) of the cone. 'l' is the slant height of the cone.

It is obtained by the area of the segment when the cone is opened / stretched along the slope.

Surface area is obtained by : pi*r the length of the curve Multiplied by the slope 'l'.

So the surface area = pi * r * l Sq. Units

Total Surface Area: pi * r * (r+l) Sq. Units (Surface area + the area of the base circle)

Answer 4

You can do it using integral calculus. Another solution is to observe that if you "unfold" a right cone you get a circular sector whose radius r in the radius of the cone basis and whose circular length g is the length of the line segment which generates the cone. So, the surface area is S = Pi * r * g

Answer 5

Start with a circle with radius R which we know has a circumference of 2*pi*R and an area pi*R^2.

Now, cut out a slice from the circle as though it was a piece of pie, but the slice makes an angle theta between two radii.

The area of the slice we find by proportion, we multiple the fraction of the circle angle by the area of the circle, a whole circle would have 2*pi radians, we cut theata/(2*pi), so:

Area of Slice = theata/(2*pi) * total area of circle
= theata/(2*pi) * pi*R^2
= (theata/2)*R^2

Connect the two ends of the circle to form a cone.

Notice that the Radius of the Circle now becomes the slant length of the Cone, so later we will rename this capital R to S for better understanding.

We now can have our first draft formula for the cone:

Surface Area of Cone = Area of Circle - Area of Slice
= pi*R^2 - (theata/2)*R^2 = (pi - theata/2)*R^2

We need to get rid of theata in the equation and express the formula in terms of little r, the base of cone radius and capital R the slant height of the cone.

We know that the area of the base of the cone is pi*r^2

The lenght along the circumference of the base of the cone is
2*pi*r

The length along the circumference of the original circle is 2*pi*R but we cut out a length of theata*R out, so the length we travel from the outer edge of the cut circle to the end to end is:
2*pi*R - theat*R = (2*pi - theata)*R = circumferance of base of cone

circumference of base of cone is now expresse in two ways

2*pi*r = (2*pi - theata)*R, use algebra to solve for theata in terms of R and r:

theata = (2*pi/R)*(R-r) ,now plug this in for theata for our 1st cone surface area formula

= (pi - theata/2)*R^2 = pi*r*R , now use S for R and we have
Surface area of cone = pi*r*S, where r is the radius of the base of the cone and S is the slant height.

Answer 6

consider a cone of height h radius r and a 2-d angke between its 2 vertices to b 2*theta and than integrate a small circle from the centre with repect to theta which is the only constant term.u will get the answer something like this
AREA=pi*r^2*cosec theta which can be simplified as pi*r *l

Answer 7

i think the proof isa very simple one.try this

let the slant height of the cone be 'l' and the eadius be 'r'
when the cone is openedout it will be a sector of the circle with radius 'l' and length of the arc=2*pi*r
area of the sector=1/2*l*r=1/2*2*pi*r*l=pi*r*l

Answer 8

it is 3.14(pie)rl
well am nt very sure about the derivation but thr was smthing like 3 cones fill a cylinder so maybe.....

Answer 9

well formula for the surface area of the cone is (pie * radius * slant height).. well derivation...................

Answer 10

dont go on deriving just get used to use it.