The surface area of a prism is the sum of the areas of all the faces.
Finding surface area using the net
To find the surface area of a prism, we need to determine the kinds of areas we need to add together.
Exploration
Consider this cube:
From this angle we can see three square faces with side length $4$4, and the area of these faces will contribute to the surface area. But we also need to consider the faces we can't see from this view.
By drawing the net of the cube we can see all the faces at once:
Now we know that the surface is made up of six identical square faces, and finding the surface area of the cube is the same as finding the area of a square face and multiplying that by $6$6:
$A=6\times4^2$A=6×42$=$=$96$96
Using the net is useful for seeing exactly what areas need to be added together, but it isn't always this easy to find.
Finding surface area using base dimensions
Another way to calculate the surface area of a prism is to calculate all the areas from the dimensions of the prism, without worrying about the exact area of each face.
Since prisms always have two identical base faces and the rest of the faces are rectangles connecting the two bases, we can accurately determine the dimensions of all the faces of a prism from just the dimensions of the base and the height of the prism.
In fact, we can think of all the rectangular faces joining the base faces as a single rectangle that wraps around the prism. One dimension of this rectangle must be the height of the prism. The other dimension of this rectangle will be the perimeter of the base.
Exploration
This rectangular prism has dimensions of $8$8, $7$7 and $5$5.
We choose the top and bottom faces to be the bases, and they each have areas of $8\times7=56$8×7=56.
To find the area of the rectangular faces joining the base faces, we multiply the height of the prism by the perimeter of one of the bases.
With two sides of length $8$8 and two sides of length $7$7, the base has a perimeter of $8+7+8+7=30$8+7+8+7=30, and multiplying by the height gives us the area $5\times30=150$5×30=150.
Adding this area to two copies of the base area tells us the total surface area for the prism:
$A=2\times56+150$A=2×56+150$=$=$262$262.
We could instead find the area of each of the six rectangles and add them together, but using the perimeter can make some calculations faster.
Practice questions
Question 1
Consider the following rectangular prism with a width, length and height of $5$5 m, $7$7 m and $15$15 m respectively. Find the surface area.
A rectangular prism consists of six rectangular faces, with each pair of opposite faces equal. The front face is a vertical rectangle. Opposite to the front face, the back face is labeled $5$5 m in width and $15$15 m in height. The top face is labeled $7$7 m in length, extending backward. Dashed lines indicate hidden edges along the back and bottom face of the prism. All visible edges are outlined with solid lines.
The curved surface area of a cylinder
By "unwrapping" the cylinder we can treat the curved surface as a rectangle, with one side length equal to the height of the cylinder, and the other the perimeter (circumference) of the base circle. This is given by $2\pi r$2πr, where $r$r is the radius.
This means the surface area of the curved part of a cylinder is $2\pi rh$2πrh, where $r$r is the radius and $h$h is the height.
We can see how the cylinder unrolls to make this rectangle in the applet below:
To find the surface area of the whole cylinder, we need to add the area of the top and bottom circles to the area of the curved part. Both of these circles have an area of $\pi r^2$πr2, so the surface area of a cylinder is:
$\text{Surface area of a cylinder}=2\pi r^2+2\pi rh$Surface area of a cylinder=2πr2+2πrh
Where $r$r is the radius and $h$h is the height of the cylinder.
Practice questions
Question 2
Consider the following cylinder.
A cylinder, outlined in black, is oriented vertically with its top circular face fully visible and its bottom circular face indicated by a dashed outline. The height is labeled $4$4 m. The radius of the bottom circular face is labeled $3$3 m.
Below the cylinder is its net represented by a horizontal rectangle and two circles. One circle is on top of the rectangle and the other circle is below the rectangle. The radius of the circle on top is labeled $3$3 m. The width of the rectangle is labeled $4$4 m.
Find the curved surface area of the cylinder to two decimal places.
Question 3
Consider the following cylinder.
A cylinder, outlined in green, is oriented horizontally with its right circular face fully visible and its left circular face indicated by a dashed outline. The height is labeled $8$8 m and is marked with a horizontal double-headed arrow across the bottom side. The radius of the circular face is labeled $4$4 m and is marked with a vertical double-headed arrow along the right circular face.
Find the curved surface area of the cylinder to two decimal places.
Using the result from part (a) or otherwise, find the total surface area of the cylinder.
Round your answer to two decimal places.
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?