As the words suggest, a surface area is the area of a surface. Here are some examples:
In construction, surface area affects planning (how much material to buy) and costs (how much to charge) in connection with such things as plasterboard, roof tiles, and paint.
In manufacturing, surface area is related to similar issues, such as the cost of making boxes or printing, or sheet metal parts.
In designing, surface area enters into calculations of wind resistance and drag in cars or aeroplanes, as well as pressure and strength of materials.
The surface area exposed to air affects how fast something cools or heats or dries out. Elephants, for example, need big ears to increase their surface area for cooling purposes.
Many objects have complex shapes to increase their surface area: the inside of your lungs, intestines, and brain, air cleaners, radiators - and towels. Others avoid flat shapes to minimise surface area and avoid drying out: pine needles and cactus for example.
Many things you buy for home use are priced by surface area - or, if not, it may be worth determining their cost per square metre to decide which is the best buy.
Prisms
A prism is a 3D shape that has two identically shaped ends joined by rectangular faces. It is named according to shape of the end faces.
We calculate the 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 as in the applet below.
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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.
The triangular prism shown below has two triangular end faces, and then three rectangular faces joining them. We could see that this shape would have a net that looks something like this:
The surface area of this shape will be the sum of the area of all the faces: the two triangles and the three rectangles.
To calculate the surface area of a prism we add up the areas of individual faces. We should take care not to miss faces or double up, but we can take shortcuts where possible (like multiplying an area by the number of identical faces).
$\text{Surface area of a prism}=\text{Sum of areas of all faces}$Surface area of a prism=Sum of areas of all faces
Practice questions
QUESTION 1
Consider the following cube with a side length equal to $6$6 cm. Find the total surface area.
QUESTION 2
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 3
Given the following triangular prism. Find the total surface area.
Question 4
The dimensions of a trapezoidal prism are given in the following diagram (not drawn to scale):
Find the surface area of the trapezoidal prism.