[FREE] Fun Math Games & Activities Packs

Always on the lookout for fun math games and activities in the classroom? Try our ready-to-go printable packs for students to complete independently or with a partner!

DOWNLOAD FREE
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 3D 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,

Surface Area image 1 US

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

Surface Area image 2 US

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

[FREE] Surface Area Worksheet (Grade 6)

[FREE] Surface Area Worksheet (Grade 6)

[FREE] Surface Area Worksheet (Grade 6)

Use this quiz to check your grade 6 students’ understanding of surface area. 10+ questions with answers covering a range of 6th grade surface area topics to identify areas of strength and support!

DOWNLOAD FREE
x
[FREE] Surface Area Worksheet (Grade 6)

[FREE] Surface Area Worksheet (Grade 6)

[FREE] Surface Area Worksheet (Grade 6)

Use this quiz to check your grade 6 students’ understanding of surface area. 10+ questions with answers covering a range of 6th grade surface area topics to identify areas of strength and support!

DOWNLOAD FREE

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,

Surface Area image 3 US

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

Surface Area table image 1

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

Surface Area table image 2


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

Surface Area table image 3

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

Surface Area table image 4

\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.

Surface Area image 5 US

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

What is surface area?

What is surface area?

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.

Surface Area image 6 US

  1. Calculate the area of each face.

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

Surface Area image 7 US

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.

Surface Area image 8 US

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.

Surface Area image 12 US

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.

Surface Area image 13 US

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.

Surface Area image 17 US

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 image 20 US
    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,
    Surface Area image 31 US
    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,
    Surface Area image 22 US
    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.

 

Surface Area image 23 US

70.2 \mathrm{~ft}^2 
GCSE Quiz False

31.2 {~ft}^2
GCSE Quiz False

15.6 {~ft}^2
GCSE Quiz True

44.25 {~ft}^2
GCSE Quiz False

\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:

 

Surface Area image 24 US

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

78 \mathrm{~ft}^2
GCSE Quiz False

148 \mathrm{~ft}^2
GCSE Quiz False

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

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.

 

Surface Area image 25 US

 

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:

 

Surface Area image 26 US

 

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

3) Calculate the surface area of the rectangular prism:

 

Surface Area image 27 US

110 \, \cfrac{2}{15} {~m}^2
GCSE Quiz True

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

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

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

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.

 

Surface Area image 28 US

 

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:

 

Surface Area image 29 US

 

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.

 

Surface Area image 30 US

54.6 \mathrm{~m}^2
GCSE Quiz False

91.65 \mathrm{~m}^2
GCSE Quiz True

66.95 \mathrm{~m}^2
GCSE Quiz False

141.05 \mathrm{~m}^2
GCSE Quiz False

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.

 

Surface Area of a Prism image 31 US

2,310 \text { units}^2
GCSE Quiz False

1,068.2 \text { units}^2
GCSE Quiz True

1,033.5 \text { units}^2
GCSE Quiz False

1,560.6 \text { units}^2
GCSE Quiz False

Calculate the area of each face.

 

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

 

Surface Area image 32 US

 

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?

 

Surface Area image 33 US

120 inches

GCSE Quiz False

9 inches

GCSE Quiz False

8 inches

GCSE Quiz False

10 inches

GCSE Quiz True

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

 

Surface Area image 34 US

 

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 

 

Surface Area image 35 US

 

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 (coming soon)

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).

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 .

Step-by-step guide: Surface area of a sphere (coming soon)

The next lessons are

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

One on one math tuition

Find out how we can help your students achieve success with our math tutoring programs.

x

[FREE] Common Core Practice Tests (Grades 3 to 6)

Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents.

40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

Download free