Math resources Geometry

Surface area

# Surface area

Here you will learn about surface area, including what it is and how to calculate it for prisms and pyramids.

Students will first learn about surface area as part of geometry in 6 th grade.

## What is surface area?

The surface area is the total area of all of the faces of a three-dimensional shape. This includes prisms and pyramids. The surface area is always recorded in square units.

Prisms are 3D shapes that have a polygonal base and rectangular faces. A rectangular prism has 6 rectangular faces, including 4 rectangular lateral faces and 2 rectangular bases.

For example,

Calculate the area of each face and then add them together for the surface area of the rectangular prism.

The surface area of the prism is the sum of the areas. Add each area twice, since each rectangle appears twice in the prism:

8+8+12+12+6+6=52 \, f t^2

You can also find the surface area by multiplying each area by 2 and then adding.

(2 \times 8)+(2 \times 12)+(2 \times 6)=52 \, f t^2

Step-by-step guide: Surface area of rectangular prism

Another type of prism is a triangular prism.

A triangular prism is made up of 5 faces, including 2 triangular bases and 3 rectangular lateral faces.

For example,

Calculate the area of each face and then add them together for the surface area of the triangular prism.

Left side
lateral face
Bottom
lateral face
Right side
lateral face
Triangular
base

\begin{aligned} A&=6 \times 6.4 \\ &=38.4 \mathrm{~mm}^2 \end{aligned}

\begin{aligned} A&=6 \times 10 \\ &=60 \mathrm{~mm}^2 \end{aligned}

\begin{aligned} A&=6 \times 6.4 \\ &=38.4 \mathrm{~mm}^2 \end{aligned}

\begin{aligned} A&=\frac{1}{2} \times 10 \times 4.2 \\ &=21 \mathrm{~mm}^2 \end{aligned}

The surface area of the prism is the sum of the areas. Add the area of the triangular base twice (or you can multiply it by 2 ), since it appears twice in the prism:

37.2+60+38.4+21+21=177.6 \mathrm{~mm}^2

Step-by-step guide: Surface area of triangular prism

Step-by-step guide: Surface area of a prism

Pyramids are another type of 3D shape. A pyramid is made up of a polygonal base and triangular lateral sides.

For example,

All lateral faces (sides) of this square pyramid are congruent.

To calculate the surface area of a pyramid, calculate the area of each face of the pyramid and then add the areas together.

\text {Area of the base }=2.5 \times 2.5=6.25 \mathrm{~cm}^2

\text {Area of a triangular face }=\cfrac{1}{2} \times 2.5 \times 4=5 \mathrm{~cm}^2

Add the area of the base and the 4 congruent triangular faces:

\text {Surface area }=6.25+5+5+5+5=6.25+(4 \times 5)=26.25 \mathrm{~cm}^2

The total surface area can also be written in one equation:

​​\begin{aligned} \text {Surface area of pyramid } & =\text {Area of base }+ \text {Areas of triangular faces } \\\\ & =2.5^2+4 \times\left(\cfrac{1}{2} \, \times 2.5 \times 4\right) \\\\ & =6.25+20 \\\\ & =26.25 \mathrm{~cm}^2 \end{aligned}

Step-by-step guide: Surface area of a pyramid

## Common Core State Standards

How does this relate to 6 th grade math?

• Grade 6 – Geometry (6.G.A.4)
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

## How to calculate the surface area of a prism

In order to calculate the surface area:

1. Calculate the area of each face.
2. Add the area of each face together.
3. Include the units.

## Surface area examples

### Example 1: surface area of a rectangular prism

Calculate the surface area of the rectangular prism.

1. Calculate the area of each face.

A rectangular prism has 6 faces, with 3 pairs of identical faces.

2Add the area of each face together.

Total surface area: 14+14+21+21+6+6=82

OR

Since opposite rectangles are always congruent, you can also use multiplication to solve:

Total surface area: 14 \times 2+21 \times 2+6 \times 2=82

3Include the units.

The measurements on this prism are in m , so the total surface area of the prism is 82 \mathrm{~m}^2.

### Example 2: surface area of a triangular prism with an equilateral triangle – using a net

Calculate the surface area of the triangular prism. The base of the prism is an equilateral triangle with a perimeter of 16.5 \, ft.

Calculate the area of each face.

Add the area of each face together.

Include the units.

### Example 3: surface area of a square-based pyramid in cm

All the lateral faces of the pyramid are congruent. Calculate the surface area.

Calculate the area of each face.

Add the area of each face together.

Include the units.

### Example 4: surface area of a rectangular prism – using a net

Calculate the lateral surface area of the rectangular prism. The base of the prism is a square and one side of the base measures 3 \, \cfrac{2}{3} inches.

Calculate the area of each face.

Add the area of each face together.

Include the units.

### Example 5: surface area of a parallelogram prism with different units

Calculate the surface area of the parallelogram prism.

Calculate the area of each face.

Add the area of each face together.

Include the units.

### Example 6: surface area of a square pyramid – word problem

Mara is making a square pyramid out of cardboard. She cut out 4 acute triangles that have a base of 5 inches and a height of 7.4 inches. How much cardboard will she need to complete the entire square pyramid?

Calculate the area of each face.

Add the area of each face together.

Include the units.

### Teaching tips for the surface area of a prism

• Make sure that students have had time to work with physical 3D models and nets before doing activities that involve finding the surface area of pyramids and prisms.

• Choose worksheets that offer a variety of question types – a mixture of showing the full pyramid or prism versus showing the net, a mixture of solving for the missing surface area versus a missing dimension, and one that includes some word problems.

### Easy mistakes to make

• Calculating volume instead of surface area
Volume and surface area are different things – volume tells us the space within the shape whereas surface area is the total area of the faces.
Particularly with rectangular prisms, it is easy to confuse these two topics if a student does not have a complete understanding of the difference.
For example,

Surface Area: Volume:
Left/right side: 4 \times 5=20 \mathrm{~ft}^2 17 \times 4 \times 5=340 \, f t^3
Front/back: 17 \times 5=85 \mathrm{~ft}^2
Top/bottom: 17 \times 4=68 \mathrm{~ft}^2
20+20+85+85+68+68=346 \mathrm{~ft}^2

• Confusing the base and the height of the prism
No matter how the prism is oriented, the height of a prism is always the side length of the lateral face that does not correspond with the base. The base is always the polygon in which the prism is named after.
For example,
This is a trapezoidal prism. It has a trapezoid as its base. Even though the trapezoids are shown in the front and back position in this image, the trapezoid is still the base and the rectangular sides are the lateral faces.

• Calculating with different units
You need to make sure all measurements are in the same units before calculating surface area.
For example,
Before calculating the surface area, the units need to all be the same. Either the cm needs to be converted to mm or the mm converted to cm.

• Confusing lateral area with total surface area
Lateral area is the area of each of the sides, and total surface area is the area of the bases plus the area of the sides. When asked to find the lateral area, be sure to only add up the area of the sides – which are always rectangles in right prisms (the types of prisms shown on this page).

Note: In oblique prisms the lateral faces are parallelograms.

### Practice surface area of a prism questions

1) The pyramid is composed of four congruent equilateral triangles. Find the surface area of the pyramid.

70.2 \mathrm{~ft}^2

31.2 {~ft}^2

15.6 {~ft}^2

44.25 {~ft}^2

\begin{aligned} \text {Surface area of pyramid }&= \text { Area of base and faces} \\ & \quad \text{ (4 congruent triangles) } \\\\ & =4 \times\left(\cfrac{1}{2} \, \times 3 \times 2.6\right) \\\\ & =4 \times 3.9 \\\\ & =15.6 \mathrm{~ft}^2 \end{aligned}

2) Calculate the surface area of the triangular prism:

\text { 150. } 5 \mathrm{~ft}^2

78 \mathrm{~ft}^2

148 \mathrm{~ft}^2

\text { 126. } 5 \mathrm{~ft}^2

You can unfold the triangular prism, and use the net to find the area of each face.

Remember that the edges in a prism always fold up together to form the prism – making their lengths equal.

The area of each triangular base:

\cfrac{1}{2} \times 5 \times 4.3=10.75

The area of each rectangular lateral face:

7 \times 5=35

If you have trouble keeping track of all the calculations, use a net:

Total surface area: 10.75+10.75+35+35+35=126.5 \mathrm{~ft}^2

3) Calculate the surface area of the rectangular prism:

110 \, \cfrac{2}{15} {~m}^2

35 \, \cfrac{7}{15} {~m}^2

29 \, \cfrac{13}{15} {~m}^2

103 \, \cfrac{11}{15} {~m}^2

You can unfold the rectangular prism, and use the net to find the area of each face.

Remember that the edges in a prism always fold up together to form the prism – making their lengths equal.

The area of each rectangular base:

\begin{aligned} & 1 \cfrac{2}{5} \, \times 8 \\\\ &= \cfrac{7}{5} \, \times \cfrac{8}{1} \\\\ &= \cfrac{56}{5} \\\\ &= 11 \, \cfrac{1}{5} \end{aligned}

The area of each rectangular lateral face:

\begin{aligned} & 4 \, \cfrac{2}{3} \, \times 8 \\\\ &= \cfrac{14}{3} \, \times \cfrac{8}{1} \\\\ &= \cfrac{112}{3} \\\\ &= 37 \, \cfrac{1}{3} \end{aligned}

\begin{aligned} & 4 \, \cfrac{2}{3} \, \times 1 \cfrac{2}{5} \\\\ &= \cfrac{14}{3} \, \times \cfrac{7}{5} \\\\ &= \cfrac{98}{15} \\\\ &= 6 \cfrac{8}{15} \end{aligned}

If you have trouble keeping track of all the calculations, use a net:

Total surface area:

\begin{aligned} & 6 \, \cfrac{8}{15} \, +6 \, \cfrac{8}{15} \, +11 \, \cfrac{1}{5} \, +11 \, \cfrac{1}{5} \, +37 \, \cfrac{1}{3} \, +37 \, \cfrac{1}{3} \\\\ & =6 \, \cfrac{8}{15} \, +6 \, \cfrac{8}{15} \, +11 \, \cfrac{5}{15} , +11 \, \cfrac{5}{15} \, +37 \, \cfrac{3}{15} \, +37 \, \cfrac{3}{15} \\\\ & =108 \, \cfrac{32}{15} \\\\ & =110 \, \cfrac{2}{15} \mathrm{~m}^2 \end{aligned}

4) Here is a net of a square pyramid. Calculate the surface area.

54.6 \mathrm{~m}^2

91.65 \mathrm{~m}^2

66.95 \mathrm{~m}^2

141.05 \mathrm{~m}^2

Calculate the area of each face.

The lateral faces are all congruent, acute triangles.

\begin{aligned} \text {Area of triangle } & =\cfrac{1}{2} \, \times 6.5 \times 3.8 \\\\ & =12.35 \end{aligned}

Since it is a square pyramid, the base is a square.

\begin{aligned} \text {Area of square } & =6.5 \times 6.5 \\\\ & =42.25 \end{aligned}

Total surface area = 12.35+12.35+12.35+12.35+42.25=91.65 \mathrm{~m}^2

5) Calculate the surface area of the prism.

2,310 \text { units}^2

1,068.2 \text { units}^2

1,033.5 \text { units}^2

1,560.6 \text { units}^2

Calculate the area of each face.

The congruent bases (front and back faces) are composed of a rectangle and a right triangle.

Total surface area = 87.5+87.5+330+220+154+189.2=1,068 .2 \text { units}^2

6) Malika was painting the hexagonal prism below. It took 140.8 \text { inches}^2 to cover the entire shape. If the area of the base is \text {10.4 inches}^2 and each side of the hexagon is 2 \text { inches} , what is the height of the prism?

120 inches

9 inches

8 inches

10 inches

You can unfold the hexagonal prism, and use the net to find the area of each face:

Total area of the bases: 10.4+10.4=20.8

Subtract the area of the bases from the total amount of paint Malika used, to see how much was used on the lateral faces:

140.8-20.8=120

The total area of the faces left is 120 \text { inches}^2.

Since the faces are congruent, the total for each face can be found by dividing by 6\text{:}

120 \div 6=20

Labeling the missing length as x , means the area of each face can be written as 2 \times x or 2 x .

Since each face has an area of 20 \text{ inches}^2 , the missing height can be found with the equation: 2 x=20.

Since 2 \times 10=20 , the missing height is 10 inches.

## Surface area FAQs

What is a cuboid?

A cuboid is a prism with a rectangular base and rectangular lateral sides. It is also known as a rectangular prism.

Are there surface area formulas for prisms?

Some shapes do have a general formula that you can use. For example, the surface area of a rectangular prism uses the formula 2 \: (l b+b h+l h) . There are other formulas, but for all prisms, the general formula is \text {area of } 2 \text { bases }+ \text {area of all lateral faces} .

How do you calculate the surface area of a cube?

Since all the faces have the same area, find the area of the square base and multiply it by 6 .

Step-by-step guide: Surface area of a cube

How do you calculate the surface area of a cylinder?

The surface area of a cylinder is the area of a circle (the two congruent bases) plus the the curved surface area (2 \pi r h). . This will give you the surface area of the cylinder.

Step-by-step guide: Surface area of a cylinder

How do you calculate the surface area of a sphere?

To find the curved surface area, square the radius of the sphere and multiply it by 4 \pi . This will give you the surface area of the sphere.

Step-by-step guide: Surface area of a sphere

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