So we know that surface area is the total area of all the faces on a three-dimensional object. We have looked at the surface area of prisms and cylinders and spheres.
A composite solid is made up of a combination of other solids, like the examples below:
To find the surface area of composite solids we need to be able to visualise the different shapes that make up the various surfaces. Once we have identified the different faces and shapes, calculate the areas of each face and add them up separately.
We should make sure to subtract faces which are not on the surface, like the circle where the cylinder sits on the rectangular prism in the middle image above.
Another common mistake is to forget to add the area of faces that are not visible in the diagram, such as faces at the back or on the bottom.
Surface area of a part of a cylinder
A closed cylinder includes all faces to make an object that would be 'watertight'.
For some problems, such as painting the outside of a section of water pipe, we might not need the total surface area. In this case, we would calculate the area of the curved face without the two circular ends:
$SA=2\pi r\times h$SA=2πr×h.
The following example looks at what happens when we cut a solid, cylindrical object to make a half-cylinder.
Worked example
Example 1
A cylindrical wooden log is split in half, lengthwise. If the unsplit log has a diameter of $20$20 cm and a length of $70$70 cm, calculate the surface area of one piece of the split log.
Think: Splitting the log creates two surfaces, a rectangle and a half cylinder.
Do: Calculate the area of the rectangle.
| $A$A | $=$= | $\text{diameter }\times\text{length }$diameter ×length |
| $=$= | $20\times70$20×70 | |
| $=$= | $1400$1400 |
Now calculate the area of half a cylinder.
| $A$A | $=$= | $\frac{1}{2}\times(2\pi r^2+2\pi rh)$12×(2πr2+2πrh) |
| $=$= | $\pi\times10^2+\pi\times10\times70$π×102+π×10×70 | |
| $=$= | $800\pi$800π | |
| $\approx$≈ | $2513.27$2513.27 |
The total surface area will be the sum of these areas:
| $SA$SA | $=$= | $1400+800\pi$1400+800π |
|
| $=$= | $3913.27$3913.27 cm2 |
To two decimal places |
Surface area of a hemisphere
A hemisphere is half of a sphere. The surface of the hemisphere is made up of the curved dome and a flat base.
The dome has an area equal to half of the surface area of the sphere, and the base has the area of a circle with the same radius as the sphere.
Worked example
Example 2
Calculate the surface area of the hemisphere below.
Think: The hemisphere has a flat, circular base that has an area given by $A=\pi r^2$A=πr2. The curved dome of the hemisphere is half of the surface area of a complete sphere so we can use the formula $A=1/2\times4\pi r^2=2\pi r^2$A=1/2×4πr2=2πr2. The total surface area will be the sum of the area of the base and the area of the dome.
Do: The radius of this sphere is $10$10 units. Let's substitute these values into the formula:
| $SA$SA | $=$= | $\pi r^2+2\pi r^2$πr2+2πr2 |
The area of the base plus the area of the dome |
| $=$= | $\pi\times10^2+2\pi\times10^2$π×102+2π×102 |
Substitute the values into the formula |
|
| $=$= | $300\pi$300π |
Evaluate the formula without rounding until the end |
|
| $=$= | $942.48$942.48 units2 |
To two decimal places |
Practice questions
Question 1
The following hemisphere has a radius of $11$11 units. Find the total surface area.
Round your answer to three decimal places.
Question 2
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.
Question 3
Find the surface area of the figure shown, rounded to two decimal places.
Question 4
A solid consists of a cylinder (of radius $5$5mm) attached to a rectangular prism on one of its faces. We wish to find the surface area of the entire solid. Note that an area is called 'exposed' if it is not covered by the other object.
What is the exposed surface area of the rectangular prism? Give your answer correct to two decimal places.
What is the exposed surface area of the cylindrical top piece? Give your answer correct to two decimal places.
(The cylinder has radius $5$5mm.)
Hence what is the total surface area (correct to two decimal places)?
Question 5
We wish to find the surface area of the shape given.
First find the area of the circular base, leaving your answer in exact form.
Now find the curved surface area of the cylinder, leaving your answer in exact form.
Finally, calculate the curved surface area of the hemisphere, leaving your answer in exact form.
Hence find the total surface area of the shape, correct to two decimal places.