11.06 Surface area of prisms and cylinders
Lateral surface area
Lateral surface area of prisms
To create a prism we start with an end face. This could be a familiar shape like a right-triangle, a square, a trapezoid or a pentagon. Or we could use some kind of irregular polygon with many different side lengths.
Our prism will have two identical end faces. The two end faces are joined together by the lateral faces.
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| A quadrilateral prism, with two end faces and four lateral faces. |
The lateral area of a prism is the total surface area of all the lateral faces. For any prism, the lateral faces will always be rectangles, so we can find the area of each lateral face using the formula for the area of a rectangle.
Each lateral face of a prism is a rectangle whose area $A$A is given by $A=bh$A=bh, where
- $b$b is the base
- $h$h is the height
If we sum the area of each lateral face of a prism, the total is the lateral area of the prism.
Worked example
Question 1
Find the lateral area of the following triangular prism.
Think: The end face is a triangle with three edges, so there are three lateral faces, one for each edge of the triangle. All three lateral faces will have the same height of $12$12 m, equal to the length of the prism.
Explore the applet below to see how the prism is made up of two triangular end faces and three rectangular lateral faces.
Do: Since each lateral face is a rectangle, we can use the formula for the area of a rectangle $A=bh$A=bh. We will label the lateral faces $A_1$A1, $A_2$A2, and $A_3$A3.
| $A_1$A1: | $A$A | $=$= | $bh$bh |
| $=$= | $3\times12$3×12 | ||
| $=$= | $36$36 m2 | ||
| $A_2$A2: | $A$A | $=$= | $bh$bh |
| $=$= | $4\times12$4×12 | ||
| $=$= | $48$48 m2 | ||
| $A_3$A3: | $A$A | $=$= | $bh$bh |
| $=$= | $5\times12$5×12 | ||
| $=$= | $60$60 m2 | ||
| Lateral area: | $A$A | $=$= | $A_1+A_2+A_3$A1+A2+A3 |
| $=$= | $36+48+60$36+48+60 | ||
| $=$= | $144$144 m2 |
So the triangular prism has a lateral area of $144$144 m2.
Reflect: Notice that the area of each lateral face is a multiple of $12$12, the length of the prism. We can write the lateral area of the prism as $12\left(3+4+5\right)$12(3+4+5), where the sum $\left(3+4+5\right)$(3+4+5) represents the perimeter of the end face.
Practice questions
Question 2
Consider the following closed rectangular prism with length of $12$12 m, width of $8$8 m and height of $24$24 m.
How many lateral faces does this prism have if the base is the side that is $8$8 m by $12$12 m?
$1$1
A$4$4
B$2$2
C$6$6
DFind the area of the lateral face, which is the rectangle with base $12$12 m and height $24$24 m.
Find the area of the lateral face, which is the rectangle with base $8$8 m and height $24$24 m.
Find the lateral area of the prism.
Question 3
Consider the following prism.
How many lateral faces does this prism have?
$\editable{}$
How many of the lateral faces are $5$5 cm by $7$7 cm?
$\editable{}$
Find the lateral area of the prism.
Question 4
Consider the following rectangular prism. Let the face with sides $w$w, $x$x, $y$y, and $z$z be the base.
Write an expression for the sum of the area of the four lateral faces.
Factor the expression from part (a).
What does the expression $w+x+y+z$w+x+y+z represent for this prism?
Perimeter of the base.
AArea of the base.
BThe surface area.
C
Lateral surface area of cylinders
The lateral area of a prism is the surface area that does not include either of its bases. For a cylinder the lateral area is the curved surface between the two circular faces, highlighted in the image below.
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| A cylinder with lateral area highlighted (left), and a net of a cylinder with the lateral area highlighted (right) |
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The net of the cylinder above shows that when we unravel the lateral area we get a rectangle. One of the dimensions on this rectangle will be the height of the cylinder.
The other dimension of the rectangle will be the circumference of the circular base, which can be found using the formula $2\pi r$2πr, where $r$r is the radius of the cylinder.
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| Dimensions of the lateral area for a cylinder | |
We can explore this further in the applet below. As the circle rolls notice how its circumference is equal to the length of the rectangle. You can drag the point on the circle, or let the animation roll the circle for you.
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For a cylinder with height $h$h and radius $r$r the lateral area will be:
$A=2\pi rh$A=2πrh
Worked example
Question 5
Find the lateral area of the cylinder below. Give your answer in square meters to two decimal places.
Think: The cylinder will unravel into a rectangle with the dimensions $2\pi r$2πr and $h$h. We can use the formula $A=2\pi r\times h$A=2πr×h to determine the lateral area.
Do: The radius of the cylinder is $3$3 m2 and the height is $7$7 m2. Let's substitute these values into the formula:
| $A$A | $=$= | $2\pi r\times h$2πr×h | (The formula for the lateral area of a cylinder) |
| $=$= | $2\pi\times3\times7$2π×3×7 | (Substitute the values into the formula) | |
| $=$= | $42\pi$42π | (Evaluate the multiplication) | |
| $=$= | $131.95$131.95 m2 | (Find the approximate answer correct to two decimal places) |
Practice questions
Question 6
Consider the following cylinder.
Find the lateral surface area of the cylinder.
Round your answer to two decimal places.
Question 7
Consider the following cylinder.
Find the lateral surface area of the cylinder to two decimal places.
question 8
The concrete water main shown below requires a protective coating before it can be used.
What is the diameter of the water main from the outer surface?
What is the lateral surface area of the exterior of the water main? Give your answer to two decimal places.
If the water main needs a protective coating on it's exterior and it costs $\$0.12$$0.12 per m2. What is the cost to coat the pipe? Give your answer to the nearestcent.
Surface area
Surface area of prisms
A prism has two end pieces which are congruent (exactly the same). It also has a number of lateral faces that join the $2$2 end pieces together.
For example, this triangular prism has $2$2 triangular end pieces and then $3$3 faces. We could see that this shape would have a net that looked like this.
The surface area of this shape will be the sum of the area of all the faces. This is the same as the total area of the net.
$2$2 triangle pieces and $3$3 rectangular pieces
We know how to find the area of a triangle, so these pieces will be easy. The three rectangles have dimensions equal to the lengths of the sides of the triangle and width equal to the height of the prism.
Exploration
Have a look at this interactive to see how to unfold prisms.
So when needing to calculate the surface area (SA) of a prism you need to add up the areas of individual faces. Take care not to miss faces or double up and look for clever methods too, like 2 faces that might have the same area!
$\text{Surface Area of a Prism }=\text{Sum of areas of all faces }$Surface Area of a Prism =Sum of areas of all faces
$\text{Surface Area of a Prism }=2\times\text{(Surface area of end face)}+\text{Lateral surface area }$Surface Area of a Prism =2×(Surface area of end face)+Lateral surface area
Practice questions
Question 9
Consider the following cube with a side length equal to $6$6 cm. Find the total surface area.
Question 10
Given the following triangular prism. Find the total surface area.
Question 11
Find the surface area of this prism.
Surface area of cylinders
Like finding the surface area of a prism the surface area of a cylinder uses a similar process.
Let's see how it would look if we open up a cylinder to view its net.
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?
This means that we can break down the total surface area of a cylinder in the following way:
| $\text{Surface Area of a Cylinder }$Surface Area of a Cylinder | $=$= | $2\times\text{(Area of circular base) }+\text{Lateral Surface Area }$2×(Area of circular base) +Lateral Surface Area |
| $=$= | $2\times\pi r^2+2\pi rh$2×πr2+2πrh | |
| $=$= | $2\pi r^2+2\pi rh$2πr2+2πrh |
Practice questions
Question 12
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?
Question 13
Find the exact surface area of a cylinder with diameter $6$6 cm and height $21$21 cm by leaving your answer in terms of $\pi$π.
Question 14
Consider the solid pictured and answer the following:
What is the external surface area of the curved surface?
Give your answer to the nearest two decimal places.
What is the total surface area of the two end pieces?
Give your answer to the nearest two decimal places.
What is the internal surface area?
Give your answer to the nearest two decimal places.
Hence what is the total surface area?
Give your answer to the nearest two decimal places.