Why does this formula work for finding the surface area of a cylinder?

Question

In math class today, we were discussing how to find the surface area of a cylinder. Then, someone figured out that you could use the formula (radius+height)(diameter). We tested the formula and it worked for every problem we had.

Why does this work or is it just a loophole?

Answer 1

Here is the proof: 

 

Surface area = 2 * area of the circular top and bottom + circumference * height

 

                   = 2 * pi * radius^2 + 2*pi*radius * height

 

                   = 2*pi*radius * ( radius + height)

 

                   = pi * diameter * (radius + height)

 

 

 

      

 

Answer 2

The total surface area of a cylinder includes top & bottom (each is a circle) + the Lateral Area.

The Lateral area = Circumference times height; that is, (2•pi•r)h.

The top circle has area pi•r²

Therefore T.S.A. = 2•pi•r²  + (2•pi•r)h

Factoring, we get 2•pi•r(r+h) = Circumference times (radius+height)

 

Answer 3

Yes, this formula does work ( you forgot to multiply by pi though). It is basically a simplified version of the 2 equations that you would generally use to calculate the surface area of a cylinder.

 

To find the surface area of a cylinder, we have 2 components: the 2 circular faces, and the curved rectangle.

The area of the 2 faces is given by 2(pi)r^2

The area of the rest of the cylinder is just a rectangle the height equal to the height of the cylinder and width equal to the circumference:

(h)(D)(pi)

 

Adding these two together gives you the total surface area:

2(pi)r^2 + h(D)(pi)

 

Lets rewrite it to achieve the equation your classmate suggested.

2(pi)r^2 + h(D)(pi), change the D to 2r:

2(pi)r^2 + h(2r)(pi), factor out common terms (2r(pi)):

2r(pi)(r + h), change the 2r back to D:

D(pi)(r + h)

There it is