A cylinder is a 3D shape that has two identical circle ends joined by a single rectangular face. It is very similar to a prism, but it has a curved surface and edges.
Surface area of a prism
We calculate the surface area of a prism by summing up the areas of the faces. In order to keep track of all the faces (and to recognise which ones have identical areas) it can be useful to visualise the net of the prism, like with the applet below.
Three-dimensional objects can be described as open or closed. For a prism, being open means that one of the faces is missing - like a box without a lid. A closed object is just one that isn't open.
Practice questions
QUESTION 1
A birthday gift is placed inside the box shown.
What is the minimum amount of wrapping paper needed to wrap this gift? (Assume the box is in the shape of a rectangular prism)
A rectangular prism is shaded blue with a pink ribbon wrapped around it, featuring a bow on top. The prism's dimensions are clearly labeled: $5$5 cm for the length, $12$12 cm for the width, and $14$14 cm for the height, with scale lines indicating these measurements.
QUESTION 2
Find the surface area of the triangular prism shown.
QUESTION 3
A swimming pool has the shape of a trapezoidal prism $14$14 metres long and $6$6 metres wide. The depth of the water ranges from $1.2$1.2 metres to $2.5$2.5 metres, as shown in the figure.
Calculate the area inside the pool that is to be tiled (assuming that the top of the pool will not be tiled).
Give your answer to the nearest $0.1$0.1 m2.
Surface area of a cylinder
Although a cylinder is not a prism (because it has a curved surface between the circular ends) we can still use its net to help us develop a rule for finding its surface area.
This applet shows how we can unravel a cylinder to view its net.
When the net is unfolded, the curved surface flattens out into a rectangle without changing its area. We can see that there are three parts to a cylinder's surface area - two circles and a rectangle.
| Surface area of a cylinder | $=$= | area of $2$2 circular ends + area of $1$1 rectangular piece |
| $=$= | $\left(2\times\pi r^2\right)+\left(L\times W\right)$(2×πr2)+(L×W) |
We know that the length of the rectangle is the height, $h$h, of the cylinder. By rotating the circle on top of the rectangle, we can see that the circumference of the circle is equal to the width of the rectangular piece! The circumference of the circle is given by $2\pi r$2πr, so we have:
$A=2\pi r^2+2\pi rh$A=2πr2+2πrh
An open cylinder is one which is missing either one or both of the circular faces - like a jar with no lid or a plastic tube. Notice that since the formula above includes the area of two circular faces it only applies for closed cylinders and will need to be adjusted for open cylinders.
Practice Questions
QUESTION 4
Consider the cylinder shown in the diagram below.
Find the surface area of the cylinder in square centimetres.
Round your answer to one decimal place.
Use your answer from part (a) to find the surface area of the cylinder in square millimetres?
Question 5
Find the surface area of the brickwork for this silo. Assume that there is a brick roof and no floor.
Give your answer correct to two decimal places.
Volume of prisms and cylinders
Both prisms and cylinders are 3D shapes that have two identically shaped ends joined by rectangular faces. The 'end' of a prism is also called the base or the cross-section, and the perpendicular distance between the ends is called the height or sometimes the length or depth of the prism. All prisms and cylinders have a uniform cross-section, meaning the cross-section remains the same size and shape across the entire 'height' of the prism.
The volume of any prism is measured in cubed units and is given by:
Volume of a prism $=$= area of end $\times$× height
This same formula applies to all prisms, even when the ends are an irregular shape. In fact, it applies to any 3D object that has a uniform cross section. In particular, if the cross-section is a circle then we have a cylinder, and so the volume of a cylinder is also
Volume of a cylinder $=$= area of end $\times$× height
| Volume | $=$= | area of end $\times$× height |
Practice questions
Question 6
A box is $1$1 metres long, $20$20 centimetres high and $30$30 centimetres wide.
Determine the volume of the box in cubic centimetres.
Question 7
Find the volume of the cylinder shown.
Round your answer to two decimal places.