6.05 Surface area of cylinders
Now that we have worked with finding the surface area of rectangular prisms, we can work to find the surface area of cylinders.
Exploration
Using the applet below, see how we can unfold a cylinder. Consider the following questions.
- What type of shape did the outside of the cylinder become when it was unfolded?
- How many circles are in the net?
- Why does the last stage show us that the base circle can roll across the rectangle?
- What is the relationship between the circles and the rectangle?
Notice that when the curved surface is unfolded, it becomes a rectangle. By rotating the circle on top of the rectangle, can you see how the circumference of the circle is equal to the length of the rectangular piece?
From this applet we can see that:
- Surface area of a cylinder is made up of two circles and one rectangle
- The radius of the circles is $r$r
- The width of the rectangle is $h$h
- The length of the rectangle is the circumference of the circle which is $2\pi r$2πr
This means that we can break down the total surface area of a cylinder in the following way:
| S.A. of a Cylinder | $=$= | Area of $2$2 circles + Area of $1$1 rectangle |
From net |
| $=$= | $2\times\pi r^2+lw$2×πr2+lw |
Using our known formulas |
|
| $=$= | $2\pi r^2+\left(2\pi r\right)\times w$2πr2+(2πr)×w |
The length of the rectangle is the circumference of the circle |
|
| $=$= | $2\pi r^2+2\pi rh$2πr2+2πrh |
The width of the rectangle is the height of the cylinder |
$\text{Surface Area of a Cylinder }=2\pi r^2+2\pi rh$Surface Area of a Cylinder =2πr2+2πrh
Practice questions
QUESTION 1
Consider the following cylinder with a height of $35$35 cm and base radius of $10$10 cm. Find the surface area of the cylinder.
A vertical cylinder with a dashed circle at the bottom indicating the base, which is hidden from view. A solid line circles the top, representing the visible base. Two measurements are shown: a horizontal line from the center of the circle to the circumference measuring $10$10cm above the top base, implying the radius, and a vertical dimension line adjacent to the right side of the cylinder measuring $35$35cm indicating the height.
Round your answer to two decimal places.
Question 2
Find the surface area of the cylinder shown.
Give your answer to the nearest two decimal places.
QUESTION 3 (extension)
A cylindrical can of radius $7$7 cm and height $10$10 cm is open at one end. What is the external surface area of the can correct to two decimal places?