10.06 Total surface area

Surface area is the total area of all the faces on a 3D object.

\text{Surface area of a prism} = \text{Sum of areas of all faces}

A prism has two parallel end pieces which are congruent (exactly the same). It also has a number of faces that join the two end pieces together.

For example, this triangular prism has 2 triangular end pieces and then 3 faces. This shape would have a net that looks like the following.

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 containing 2 triangle pieces and 3 rectangle pieces.

The three rectangles have dimensions equal to the lengths of the sides of the triangle and width equal to the height of the prism.

Examples

Example 1

Given the following triangular prism. Find the total surface area.

Worked Solution

Create a strategy

To find the surface area of the triangular prism, we want to sum up the areas of each face.

Apply the idea

\displaystyle \text{Area of triangles}	\displaystyle =	\displaystyle \dfrac{1}{2}\times 8 \times 6 \times 2	Multiply the area of each triangle by 2
	\displaystyle =	\displaystyle 48	Evaluate
\displaystyle \text{Front rectangle}	\displaystyle =	\displaystyle 10 \times 18	Multiply the base by the length
	\displaystyle =	\displaystyle 180	Evaluate
\displaystyle \text{Back rectangle}	\displaystyle =	\displaystyle 6 \times 18	Multiply the base by the length
	\displaystyle =	\displaystyle 108	Evaluate
\displaystyle \text{Side rectangle}	\displaystyle =	\displaystyle 8 \times 18	Multiply the base by the length
	\displaystyle =	\displaystyle 144	Evaluate
\displaystyle \text{Surface Area}	\displaystyle =	\displaystyle 48 + 180 + 108 + 144	Add the areas of each face
	\displaystyle =	\displaystyle 480\text{ cm}^2	Evaluate

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The surface of a cylinder is made up of circular faces on the top and bottom and a rectangular face that wraps around the curved surface of the cylinder, as shown in the diagram below.

A cylinder and its corresponding net.

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, where r is the radius of the cylinder.

Dimensions of the curved face of a cylinder

Type 1 - cylinder with two closed ends

The surface area of a closed cylinder is the sum of the area of the two circular end faces and the area of the rectangle formed by unrolling the curved face. SA=2\times \pi r^2+2\pi r\times h

Type 2 - cylinder with one closed end (like a circular swimming pool) SA=\pi r^2+2\pi r\times h

Type 3 - cylinder with two open ends (like a tunnel) SA=2\pi r\times h

If we are given the diameter of the cylinder, d then we first need to calculate the radius, r=\dfrac{d}{2}, to use the surface area formula.

Examples

Example 2

Find the surface area of the cylinder shown. Give your answer to the nearest two decimal places.

Worked Solution

Create a strategy

Use the formula: \text{Surface Area} = 2 \pi r^2 + 2 \pi r h

Apply the idea

We are given: r=6 and h=10.

\displaystyle \text{Surface Area}	\displaystyle =	\displaystyle 2 \pi \times 6^2 + 2 \pi \times 6 \times 10	Substitute r=6 and h=10
	\displaystyle =	\displaystyle 603.19\text{ cm}^2	Evaluate

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Example 3

A cylindrical can of radius 7\text{ cm} and height 9\text{ cm} is open at one end. What is the external surface area of the can correct to two decimal places?

Worked Solution

Create a strategy

Use the formula: \pi r^2+2\pi rh.

Apply the idea

We are given: r=7 and h=9.

\displaystyle \text{Surface Area}	\displaystyle =	\displaystyle \pi \times (7)^2 + 2 \times \pi \times 7 \times 9	Substitute r=7 and h=9
	\displaystyle =	\displaystyle 549.78\text{ cm}^2	Evaluate

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Example 4

The diagram shows a water trough in the shape of a half cylinder.

Find the external surface area of this water trough.

Leave your answer in exact form in terms of \pi.

Worked Solution

Create a strategy

Use the formula: 2\pi rh+2\pi r^2, and divide it by 2 since the water trough in the shape of a half cylinder.

Apply the idea

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle \dfrac{2\times \pi \times 0.62 \times 2.42 + 2 \times \pi \times (0.62)^2}{2}	Substitute r=0.62 and h=2.42
	\displaystyle =	\displaystyle \dfrac{3.000\,8\pi + 0.768\,8\pi}{2}	Evaluate the product
	\displaystyle =	\displaystyle 1.884\,8\pi \text{ m}^2	Evaluate

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A pyramid can be made in the following way. Use any polygon as a base. There can be square bases, triangular bases or even hexagonal bases. Then, connect every vertex of the base to an apex point above the base, and you have a pyramid.

Square and rectangular based pyramids, are the most common you will come across in mathematics, but also in the real world.

\text{Surface area of right pyramid} = \text{Area of base} + \text{Area of triangles}

Examples

Example 5

Find the surface area of the square pyramid shown. Include all faces in your calculations.

Worked Solution

Create a strategy

Add the area of the 4 triangles and the area of the square.

Apply the idea

Area of the 4 triangles:

\displaystyle A	\displaystyle =	\displaystyle \dfrac{1}{2}\times bh \times 4	Write the formula
	\displaystyle =	\displaystyle \dfrac{1}{2}\times 4\times 10 \times 4	Substitute b=4 and h=10
	\displaystyle =	\displaystyle 80\text{ cm}^2	Evaluate

Area of the square:

\displaystyle A	\displaystyle =	\displaystyle s^2	Write the formula
	\displaystyle =	\displaystyle 4^2	Substitute s=4
	\displaystyle =	\displaystyle 16\text{ cm}^2	Evaluate

Surface area of the square pyramid:

\displaystyle \text{Surface Area}	\displaystyle =	\displaystyle 80 + 16	Add the areas of triangles and square
	\displaystyle =	\displaystyle 96\text{ cm}^2	Evaluate

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A solid three-dimensional circular object is a sphere.

Archimedes showed that the surface area of the circular component of the cylinder wrapping the circle, has area 2\pi r\times 2r=4\pi r^2 and that this area is the same value for the surface area of a sphere.

\text{Surface area of a sphere} = 4\pi r^2

Examples

Example 6

Find the surface area of the sphere shown. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the surface area formula SA=4\pi r^2.

Apply the idea

We are given r=9.

\displaystyle SA	\displaystyle =	\displaystyle 4\pi r^2	Write the formula
	\displaystyle =	\displaystyle 4\times\pi \times 9^2	Substitute r
	\displaystyle \approx	\displaystyle 1017.88 \text{ cm}^2	Evaluate and round

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Sometimes, the shape is a composite solid, it is made up of a combination of other solids. These are some examples of composite solids.

To find the surface area of composite solids, it is necessary to be able to visualise the different shapes that make up the various surfaces. Once these are identified (the different faces and shapes), then calculate the areas of each face and add them up separately.

Don't forget 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.

Examples

Example 7

Find the surface area of the house below.

Worked Solution

Create a strategy

Add all the faces together. The diagram below shows which faces are equal.

Use Pythagoras' theorem, c^2=a^2+b^2, to find the width of the roof panels.

Apply the idea

\displaystyle \text{Front and back}	\displaystyle =	\displaystyle \left(\text{rectangle}+\text{triangle}\right)\times 2	Add the areas and multiply by 2
	\displaystyle =	\displaystyle \left(6\times 5+\dfrac{1}{2}\times 5 \times 3\right)\times 2	Substitute the dimensions
	\displaystyle =	\displaystyle 75\text{ m}^2	Evaluate

\displaystyle \text{Side walls}	\displaystyle =	\displaystyle 9\times 6 \times 2	Multiply the length and width and double
	\displaystyle =	\displaystyle 108\text{ m}^2	Evaluate

\displaystyle \text{Base}	\displaystyle =	\displaystyle 5\times 9	Multiply the length and width
	\displaystyle =	\displaystyle 45\text{ m}^2	Evaluate

Using Pythagoras' theorem, we can find the width of the roof panels. On the given diagram, we can see two right-angled triangles where the shorter sides are a=3 and b=\dfrac{5}{2}=2.5.

\displaystyle c^2	\displaystyle =	\displaystyle 3^2+(2.5)^2	Use Pythagoras' theorem
\displaystyle c^2	\displaystyle =	\displaystyle 15.25	Evaluate
\displaystyle c	\displaystyle =	\displaystyle \sqrt{15.25}	Square root both sides

\displaystyle \text{Roof sides}	\displaystyle =	\displaystyle 9\times \sqrt{15.25} \times 2	Find the area of the 2 roof sides
	\displaystyle =	\displaystyle 18\sqrt{15.25}\text{ m}^2	Evaluate

\displaystyle SA	\displaystyle =	\displaystyle 75+108+45+18\sqrt{15.25}	Add all the areas
	\displaystyle =	\displaystyle 298.29\text{ m}^2	Evaluate

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Example 8

Find the surface area of the given solid.

a

What is the surface area of the faces as seen from the top view?

Worked Solution

Create a strategy

We can use the formula: \text{Area} = l \times w to find the area of each faces, and add them together.

Apply the idea

Viewed from the top we have two rectangles, we are given: l=48,\, l=48,\,w=19 and w=14.

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 48 \times 19 + 48 \times 14	Substitute the values of lengths and width
	\displaystyle =	\displaystyle 912 + 672	Evaluate the multiplication
	\displaystyle =	\displaystyle 1584\text{ cm}^2	Evaluate

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b

What is the surface area of the faces as seen from the left side view?

Worked Solution

Create a strategy

We can use the formula: \text{Area} = l \times w to find the area of the faces, and add them together.

Apply the idea

We can consider the surface in view from the left side as either two rectangles laid next to each other. We are given: l=19,\,l=13,\,w=14 and w=19.

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 19 \times 14 + 13 \times 19	Substitute the values of lengths and widths
	\displaystyle =	\displaystyle 266 + 247	Evaluate the multiplication
	\displaystyle =	\displaystyle 513\text{ cm}^2	Evaluate

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c

What is the surface area of the faces as seen from the front view?

Worked Solution

Create a strategy

We can use the formula: \text{Area} = l \times w to find the area of each faces, and add them together.

Apply the idea

From the front we see two rectangles, we are given: l=48,\,l=48,\,w=13 and w=6.

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 48 \times 13 + 48 \times 6	Substitute the values of lengths and widths
	\displaystyle =	\displaystyle 624 + 288	Evaluate the multiplication
	\displaystyle =	\displaystyle 912\text{ cm}^2	Evaluate

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d

Therefore, what is the total surface area, including all faces of the solid?

Worked Solution

Create a strategy

Multiply all the surface areas from part (a), part (b), and part (c) by 2 and then add altogether.

Apply the idea

\displaystyle \text{Total surface area}	\displaystyle =	\displaystyle 2\times 1584 + 2 \times 513 + 2 \times 912	Multiply the surface areas by 2
	\displaystyle =	\displaystyle 3168 + 1026 + 1824	Evaluate the multiplication
	\displaystyle =	\displaystyle 6018\text{ cm}^2	Evaluate

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