Question

Explain how to derive the surface area formula of a cylinder: 2πr^2 + 2πrh

Answer 1

Step-by-step explanation:

Look at the picture.

We have the rectangle

$2\pi r\times H$

and two circles with radius r.

The area of a rectangle (lateral area):

$L.A.=(2\pi r)(H)=2\pi rH$

The area of a circle (area of a base):

$B=\pi r^2$

The surface area of a cylinder:

$S.A.=2B+L.A.$

Substitute:

$S.A.=2\pi r^2+2\pi rH$

Answer 2

The surface area of a cylinder can be found by breaking it down into three parts:

The two circles that make up the ends of the cylinder.
The side of the cylinder, which when "unrolled" is a rectangle
In the figure above, drag the orange dot to the left as far as it will go. You can see that the cylinder is made up of two circular disks and a rectangle that is like the label unrolled off a soup can.

The area of each end disk can be found from the radius r of the circle.
The area of a circle is πr2, so the combined area of the two disks is twice that, or2πr2.
(See Area of a circle).
The area of the rectangle is the width times height.
The width is the height h of the cylinder, and the length is the distance around the end circles. This is the circumference of the circle and is 2πr. Thus the rectangle's area is 2πr × h.
Combining these parts we get the final formula:
area = 2 π r 2 + 2 π r h
where:
π is Pi, approximately 3.142
r is the radius of the cylinder
h height of the cylinder


By factoring 2πr from each term we can simplify this to
area = 2 π r ( r + h )


Hope that helped. Make sure to hit the thanks if it did.. or reply with any help.