Surface Area of a Prism - Math Steps, Examples & Questions (2024)

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Introduction

What is the surface area of a prism?

Common Core State Standards

How to calculate the surface area of a prism

Surface area of a prism examples

Example 1: surface area of a triangular prism with a right triangleExample 2: surface area of a rectangular prismExample 3: surface area of a triangular prism with a equilateral triangle - using a netExample 4: surface area of a rectangular prism - using a netExample 5: surface area of a parallelogram prism with different unitsExample 6: surface area word problem - missing height

Teaching tips for the surface area of a prism

Easy mistakes to make

Related surface area lessons

Practice surface area of a prism questions

Surface area of a prism FAQs

Next lessons

Still stuck?

Math resources Geometry Surface area

Surface area of a prism

Here you will learn about the surface area of a prism, including what it is and how to calculate it.

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

What is the surface area of a prism?

The surface area of a prism is the total area of all of the faces of the prism. Prisms are 3D shapes made up of flat faces, including polygonal bases and rectangle faces.

One type of prism is a rectangular prism.

A rectangular prism is made up of 6 rectangular faces, including 2 rectangular bases and 4 rectangular lateral faces.

For example,

Here is a rectangular prism and its net.

Surface Area of a Prism - Math Steps, Examples & Questions (3)

It has 3 pairs of congruent faces, since the opposite faces are the same.

Surface Area of a Prism - Math Steps, Examples & Questions (4)

To calculate the surface area of the rectangular prism, calculate the area of each face and then add them together.

FaceArea

Surface Area of a Prism - Math Steps, Examples & Questions (5)
A=5 \times 8=40 \, cm^2

Surface Area of a Prism - Math Steps, Examples & Questions (6)
A=3 \times 8=24 \, cm^2

Surface Area of a Prism - Math Steps, Examples & Questions (7)
A=5 \times 3=15 \, cm^2

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

40+40+24+24+15+15=158 \mathrm{~cm}^2

The surface area of the rectangular prism is equal to 158 \mathrm{~cm}^2 .

Note: Surface area is measured in square units. For example, \mathrm{mm}^2, \mathrm{~cm}^2, \mathrm{~m}^2 , etc.

Surface Area of a Prism - Math Steps, Examples & Questions (8)

[FREE] Surface Area Check for Understanding Quiz (Grade 6)

Surface Area of a Prism - Math Steps, Examples & Questions (9)

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!

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Surface Area of a Prism - Math Steps, Examples & Questions (10)

[FREE] Surface Area Check for Understanding Quiz (Grade 6)

Surface Area of a Prism - Math Steps, Examples & Questions (11)

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 triangular bases and 3 rectangular lateral faces.

For example,

Here is a triangular prism and its net.

Surface Area of a Prism - Math Steps, Examples & Questions (12)

The base of this triangular prism is an isosceles triangle – two of the side lengths are equal.

Remember that the edges in a prism are always equal, so if you were to fold up the net, the 6.5 \mathrm{~mm} side of the triangle would combine to form an edge with each corresponding rectangle – making their lengths equal.

Surface Area of a Prism - Math Steps, Examples & Questions (13)

To calculate the surface area of the triangular prism, calculate the area of each face and then add them together.

FaceArea

Surface Area of a Prism - Math Steps, Examples & Questions (14)
A=5 \times 6.5=32.5 \mathrm{~mm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (15)
A=5 \times 10=50 \mathrm{~mm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (16)
A=5 \times 6.5=32.5 \mathrm{~mm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (17)
A=\cfrac{1}{2} \times 10 \times 4.2=21 \mathrm{~mm}^2

The surface area of the prism is the sum of the areas. Add the area of the triangular base twice, since it appears twice in the prism:

32.5+50+32.5+21+21=157 \mathrm{~mm}^2

The surface area of the triangular prism is equal to 157 \mathrm{~mm}^2.

Surface area is measured in square units. For example, \mathrm{mm}^2, \mathrm{~cm}^2, \mathrm{~m}^2 , etc.

What is the surface area of a prism?

Surface Area of a Prism - Math Steps, Examples & Questions (18)

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 work out the surface area of a prism:

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

Surface area of a prism examples

Example 1: surface area of a triangular prism with a right triangle

Calculate the surface area of the triangular prism.

Surface Area of a Prism - Math Steps, Examples & Questions (19)
  1. Calculate the area of each face.
Surface Area of a Prism - Math Steps, Examples & Questions (20)

The area of the front of the prism is \cfrac{1}{2} \, \times 4 \times 3= 6 \mathrm{~cm}^{2}.

The back face is the same as the front face so the area of the back face is also 6 \mathrm{~cm}^{2}.

Surface Area of a Prism - Math Steps, Examples & Questions (21)

The area of the bottom is 4 \times 2= 8 \mathrm{~cm}^2.

Surface Area of a Prism - Math Steps, Examples & Questions (22)

The area of the left side is 2 \times 3=6 \mathrm{~cm}^2.

Surface Area of a Prism - Math Steps, Examples & Questions (23)

The area of the top side is 2 \times 5=10 \mathrm{~cm}^2.

It will make our working clearer if we use a table:

FaceArea
Front \cfrac{1}{2} \times 4 \times 3=6
Back 6
Bottom 4 \times 2=8
Left side 2 \times 3=6
Top 2 \times 5=10

2Add the area of each face together.

Total surface area: 6+6+8+6+10= 36

3Include the units.

The measurements on this prism are in cm , so the total surface area of the prism is 36 \mathrm{~cm}^2 .

Example 2: surface area of a rectangular prism

Calculate the surface area of the rectangular prism.

Surface Area of a Prism - Math Steps, Examples & Questions (24)

Calculate the area of each face.

Add the area of each face together.

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

Include the units.

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

Example 3: 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 \mathrm{~ft} .

Surface Area of a Prism - Math Steps, Examples & Questions (26)

Calculate the area of each face.

Add the area of each face together.

Total surface area: 13.2+13.2+55+55+55= 191.4

Include the units.

The measurements on this prism are in ft , so the total surface area of the prism is 191.4 \mathrm{~ft}^2 .

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} \text { inches} .

Surface Area of a Prism - Math Steps, Examples & Questions (30)

Calculate the area of each face.

Add the area of each face together.

Include the units.

The measurements on this prism are in inches, so the total lateral surface area of the prism is 143 \cfrac{11}{15} \text { inches }^2 .

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

Calculate the surface area of the parallelogram prism.

Surface Area of a Prism - Math Steps, Examples & Questions (34)

Calculate the area of each face.

Add the area of each face together.

Total surface area: 0.48+0.48+1.8+1.8+0.75+0.75= 6.06

Include the units.

The measurements that we have used are in m , so the surface area of the prism is 6.06 \mathrm{~m}^2 .

Example 6: surface area word problem – missing height

Surface Area of a Prism - Math Steps, Examples & Questions (37)

Ginny painted an area of 264 \mathrm{~ft}^2 on the rectangular building shown above. She painted all sides, except the bottom. How many feet long is the building

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 asking them to solve with surface area.
  • Choose worksheets that offer a variety of question types – a mixture of the prism shown or the net, a mixture of solving for the missing surface area or height/length/width, and 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 of a Prism - Math Steps, Examples & Questions (40)
    Surface Area: Volume:
    Left/right side: 4 \times 5=20 \mathrm{~ft}^2 17 \times 4 \times 5=340 \mathrm{~ft}^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 rectangular 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 of a Prism - Math Steps, Examples & Questions (41)
    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 of a Prism - Math Steps, Examples & Questions (42)
    Before calculating the surface area, all the units need to 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 lateral area, be sure to only add up the area of the sides – which are always rectangles in a prism.

Related surface area lessons

  • Surface area
  • Surface area of a rectangular prism
  • Surface of a cylinder
  • Surface area of a triangular prism
  • Surface area of a pyramid
  • Surface area of a cone
  • Surface area of a sphere
  • Surface area of a cube
  • Surface area of a hemisphere

Practice surface area of a prism questions

1. Calculate the surface area of the triangular prism:

Surface Area of a Prism - Math Steps, Examples & Questions (43)

288 \mathrm{~cm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (44)

336 \mathrm{~cm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (45)

384 \mathrm{~cm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (46)

408 \mathrm{~cm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (47)

Calculate the area of each face.

Surface Area of a Prism - Math Steps, Examples & Questions (48)

Total surface area = 24+24+96+72+120= 336 \mathrm{~cm}^{2}

2. Calculate the surface area of the rectangular prism:

Surface Area of a Prism - Math Steps, Examples & Questions (49)

154 \mathrm{~mm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (50)

113 \mathrm{~mm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (51)

336 \mathrm{~mm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (52)

226 \mathrm{~mm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (53)

Calculate the area of each face.

Surface Area of a Prism - Math Steps, Examples & Questions (54)

Total surface area = 14+14+77+77+22+22= 226 \mathrm{~mm}^{2}

3. Calculate the surface area of the triangular prism:

Surface Area of a Prism - Math Steps, Examples & Questions (55)

168 \mathrm{~cm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (56)

250.8 \mathrm{~cm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (57)

223.2 \mathrm{~cm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (58)

247.2 \mathrm{~cm}^2

Surface Area of a Prism - Math Steps, Examples & Questions (59)

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 of a Prism - Math Steps, Examples & Questions (60)

The area of each triangular base:

\cfrac{1}{2} \times 4 \times 6= 12

The area of each rectangular lateral face:

12 \times 6.3= 75.6

12 \times 4= 48

12 \times 6.3= 75.6

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

Surface Area of a Prism - Math Steps, Examples & Questions (61)

Total surface area: 12+12+75.6+48+75.6= 223.2 \mathrm{~cm}^2

4. Calculate the surface area of the rectangular prism:

Surface Area of a Prism - Math Steps, Examples & Questions (62)

87 \, \cfrac{13}{15} \mathrm{~m}^2

Surface Area of a Prism - Math Steps, Examples & Questions (63)

36 \, \cfrac{6}{15} \; m^2

Surface Area of a Prism - Math Steps, Examples & Questions (64)

138 \, \cfrac{2}{15} \mathrm{~m}^2

Surface Area of a Prism - Math Steps, Examples & Questions (65)

42 \, \cfrac{14}{15} \mathrm{~m}^2

Surface Area of a Prism - Math Steps, Examples & Questions (66)

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 of a Prism - Math Steps, Examples & Questions (67)

The area of each rectangular base:

\begin{aligned}& 1 \cfrac{1}{5} \times 7 \\\\& =\cfrac{6}{5} \times \cfrac{7}{1} \\\\& =\cfrac{42}{5} \\\\& =8 \cfrac{2}{5}\end{aligned}

The area of each rectangular lateral face:

\begin{aligned}& 4 \cfrac{1}{3} \times 7 \\\\& =\cfrac{13}{3} \times \cfrac{7}{1} \\\\& =\cfrac{91}{3} \\\\& =30 \cfrac{1}{3}\end{aligned}

\begin{aligned}& 4 \cfrac{1}{3} \times 1 \cfrac{1}{5} \\\\& =\cfrac{13}{3} \times \cfrac{6}{5} \\\\& =\cfrac{78}{15} \\\\& =5 \cfrac{3}{15}\end{aligned}

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

Surface Area of a Prism - Math Steps, Examples & Questions (68)

Total surface area:

\begin{aligned}& 8 \cfrac{2}{5}+8 \cfrac{2}{5}+30 \cfrac{1}{3}+30 \cfrac{1}{3}+5 \cfrac{3}{15}+5 \cfrac{3}{15} \\\\& =8 \cfrac{6}{15}+8 \cfrac{6}{15}+30 \cfrac{5}{15}+30 \cfrac{5}{15}+5 \cfrac{3}{15}+5 \cfrac{3}{15}\\\\& =86 \cfrac{28}{15} \\\\& =87 \cfrac{13}{15} \mathrm{~m}^2\end{aligned}

5. Calculate the surface area of the prism:

Surface Area of a Prism - Math Steps, Examples & Questions (69)

3,570 \mathrm{~m}^2

Surface Area of a Prism - Math Steps, Examples & Questions (70)

3,210 \mathrm{~m}^2

Surface Area of a Prism - Math Steps, Examples & Questions (71)

11,700 \mathrm{~m}^2

Surface Area of a Prism - Math Steps, Examples & Questions (72)

3,330 \mathrm{~m}^2

Surface Area of a Prism - Math Steps, Examples & Questions (73)

Break the base up into a rectangle and a triangle and find the area of each:

Surface Area of a Prism - Math Steps, Examples & Questions (74)

Area A :

24 \times 30= 720

Area B :

\cfrac{1}{2} \times 24 \times 5= 60

Total area of the base: 720+60= 780

Calculate the area of each face:

Surface Area of a Prism - Math Steps, Examples & Questions (75)

Total surface area : 780+780+450+360+450+195+195= 3,210 \mathrm{~m}^{2}

6. Zahir was painting the pentagonal prism below. It took 820 \text { inches}^2 to cover the entire shape. If the area of the base is 140 \text { inches}^2 and each side of the pentagon is 9 \text { inches} , what is the height of the prism?

Surface Area of a Prism - Math Steps, Examples & Questions (76)

120 \text { inches }

Surface Area of a Prism - Math Steps, Examples & Questions (77)

440 \text { inches }

Surface Area of a Prism - Math Steps, Examples & Questions (78)

9 \text { inches }

Surface Area of a Prism - Math Steps, Examples & Questions (79)

12 \text { inches }

Surface Area of a Prism - Math Steps, Examples & Questions (80)

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

Surface Area of a Prism - Math Steps, Examples & Questions (81)

Total area of the bases: 140+140= 280

Subtract the area of the bases from the total amount of paint Zahir used, to see how much was used on the lateral faces: 820-280= 540

The total area of the faces left is 540 \text { inches}^2.

Since the 5 faces are congruent, the total for each face can be found by dividing by 5 :

540 \div 5= 108

Surface Area of a Prism - Math Steps, Examples & Questions (82)

Labeling the missing length as x , means the area of each face can be written as 9 \times x or 9 x.

Since each face has an area of 108 \text { inches }^2 , the missing height can be found with the equation: 9 x=108.

Since 9 \times 12=108 , the missing height is 12 inches.

Surface area of a prism FAQs

Is there a surface area of a triangular prism formula?

The general formula is (S1+ S+ S3) L + bh , but it is not common for students to memorize this formula. It is typically easier for students to remember to find the area of the bases and the three rectangular faces, since these dimensions often vary. Finally, add the area of the faces all together to find the total surface area.

What are some different types of prisms not shown on this page?

There are trapezoidal prisms, which have a trapezoid base. There are also hexagonal prisms, which have a hexagon as a base.

The next lessons are

  • Pythagorean theorem
  • Congruence and similarity
  • Transformations

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Surface Area of a Prism - Math Steps, Examples & Questions (83)

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Introduction

What is the surface area of a prism?

Common Core State Standards

How to calculate the surface area of a prism

Surface area of a prism examples

Example 1: surface area of a triangular prism with a right triangleExample 2: surface area of a rectangular prismExample 3: surface area of a triangular prism with a equilateral triangle - using a netExample 4: surface area of a rectangular prism - using a netExample 5: surface area of a parallelogram prism with different unitsExample 6: surface area word problem - missing height

Teaching tips for the surface area of a prism

Easy mistakes to make

Related surface area lessons

Practice surface area of a prism questions

Surface area of a prism FAQs

Next lessons

Still stuck?

x

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Surface Area of a Prism - Math Steps, Examples & Questions (2024)

FAQs

Surface Area of a Prism - Math Steps, Examples & Questions? ›

Step 1: Write the given dimensions of the prism. Step 2: Substitute the given values in the formula S = (2 × Base Area) + (Base perimeter × height) where "S" is the surface area of the prism. Step 3: Now solve the equation for "Base Area by simplifying the equations".

What is surface area of prism grade 6? ›

The surface area of a prism is the sum of the areas of all its faces. A two-dimensional representation of a solid is called a net.

Which is a correct method for finding the surface area of prisms? ›

To calculate the total surface area of a prism:
  • Find the area of the two end faces.
  • Work out the area of all the rectangular faces in one of two ways: Work out the area of each rectangle separately, length × width. Multiply the perimeter of the end face by the length of the prism.
  • Sum the areas of all the faces.

What is the surface area of a prism for kids? ›

To calculate the surface area of the cuboid, we need to calculate the area of each face, and then add them together. Now that we know the area of each face, the surface area of the prism is the sum of these values. 40+24+15+40+24+15=158. 40 + 24 + 15 + 40 + 24 + 15 = 158.

How do you find surface area step by step? ›

Multiply the length and width, or c and b to find their area. Multiply this measurement by two to account for both sides. Add the three separate measurements together. Because surface area is the total area of all of the faces of an object, the final step is to add all of the individually calculated areas together.

What is the formula for a prism? ›

Formula for the volume of a prism:

Prism volume (V) = B × h, where, B is the area of the base and h is the height of the prism.

What is the rule for the surface area of a prism? ›

The formula for the surface area of a prism is SA=2B+ph, where B, again, stands for the area of the base, p represents the perimeter of the base, and h stands for the height of the prism.

What is the basic formula for surface area? ›

Variables:
Surface Area FormulaSurface Area Meaning
SA=2B+PhFind the area of each face. Add up all areas.
SA=B+12sPFind the area of each face. Add up all areas.
SA=2B+2πrhFind the area of the base, times 2, then add the areas to the areas of the rectangle, which is the circumference times the height.
2 more rows
Sep 17, 2020

What is the surface area explained easily? ›

Surface area is the amount of space covering the outside of a three-dimensional shape.

How do you find the surface area of a prism formula sheet? ›

Surface Area of Prism Formula
ShapeBase of the prismTotal Surface Area [(2 × Base Area) + (Base perimeter × height)]
Triangular PrismTriangle(a + b + c)H + bh square units
Rectangular PrismRectangle2 (lh + wh + lw) square units
Square PrismSquare[4sh + 2s2] square units
Pentagonal PrismPentagon[5ab + 5bh] square units
1 more row
Jun 4, 2024

How do you explain surface area to a child? ›

What is surface area? Surface area measures the space needed to cover the outside of a three-dimensional shape. Surface area is the sum of the areas of the individual sides of a solid shape. Surface area is measured in square units.

What is the total surface area of the prism calculator? ›

If “l” is the length, and “h” is the height and “w” is the width of the rectangular prism, the surface area is given by the formula, The surface area of a rectangular prism = 2 (lw+lh+hw) square units.

What is the formula for the total surface area of a square prism? ›

Surface area of a square prism = 4 × (s × h) = 4sh, where, s is the length of the side of the square and h is the height of the square prism.

What is the surface area of a regular prism? ›

Therefore, the lateral surface area (LSA) of a rectangular prism = 2 ( l + w ) h square units. Therefore, the surface area of a rectangular prism formula is given as, Surface Area of a rectangular prism = 2 (lh +wh + lw ) Square units.

What is the correct formula for surface area? ›

Surface Area Formula List
ShapeLateral Surface Area (LSA)Total Surface Area (TSA)
Cuboid2h(l+b)2(lb + lh + bh)
Cylinder2πrh2πr(r + h)
Coneπrlπr(l + r)
Sphere4πr24πr2
4 more rows
Feb 19, 2024

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