2.06 Surface area of composite shapes

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

Question 2

Question 3

Question 4

Question 5