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## What is the area of surfaces of prisms?

The area of surfaces of prisms is the total plane surface occupied by the sides of 3-dimensional geometrical figures that have **constant cross-sections **throughout their body. A prism has identical ends and **flat faces**.

The area of surfaces of prisms is measured in squared centimeters, meters, feet (cm^{2}, m^{2}, ft^{2}), etc.

The total surface area of a prism is the sum of twice its base area and the product of the perimeter of the base and the height of the prism.

There are many different types of prisms that obey the rules and formula mentioned above. In general, it can be said that all polygons can become prisms in 3D and hence their total surface areas can be calculated. Let us look at some examples.

### Triangular Prism

A triangular prism has 5 faces including 2 triangular faces and 3 rectangular ones.

An image of a triangular prism, StudySmarter Originals

### Rectangular Prism

A rectangular prism has 6 faces, all of which are rectangular.

An image of a rectangular prism, StudySmarter Originals

### Pentagonal Prism

A pentagonal prism has 7 faces including 2 pentagonal faces and 5 rectangular faces.

An image of a pentagonal prism, StudySmarter Originals

### Trapezoidal Prism

A trapezoidal prism has 6 faces including 2 trapezoidal faces and 4 rectangular ones.

An image of a trapezoidal prism, StudySmarter Originals

### Hexagonal Prism

A hexagonal prism has 8 faces including 2 hexagonal faces and 6 rectangular faces.

An image of a hexagonal prism, StudySmarter Originals

A cylinder is not considered a prism because it has curved surfaces, not flat ones.

## What is the method of finding the surface area of a prism?

The method which brought about the calculation of the surface area of a prism was the consideration of every side of the prism. In order to do this, we need to analyse what a simple prism consists of.

Every prism consists of two faces which are identical in both shape and dimension. We call these two faces the top and base.

An illustration of the top and base faces of a prism using a triangular prism, StudySmarter Originals

It also comprises rectangular surfaces depending on the number of sides the prism base has. For instance, a triangular base prism will have 3 other sides aside from its identical top and base. Likewise, a pentagonal base prism will have 5 other sides apart from its identical top and base, and this applies to all prisms.

An illustration of the rectangular faces of a prism using a triangular prism, StudySmarter Originals

Always remember that the sides which are different from the top and base are rectangular - this will help you in understanding the approach used in developing the formula.

Now that we know what the surfaces of a prism comprise, it is easier to calculate the total surface area of a prism. We have 2 identical sides which take the shape of the prism, and n rectangular sides - where n is the number of sides of the base.

The area of the top must surely be the same as the base area which depends on the shape of the base. So, we can say that the total surface area of both the top and base of the prism is

${A}_{B}=basearea\phantom{\rule{0ex}{0ex}}{A}_{T}=toparea\phantom{\rule{0ex}{0ex}}{A}_{TB}=Areaofbaseandtop\phantom{\rule{0ex}{0ex}}{A}_{B}={A}_{T}\phantom{\rule{0ex}{0ex}}{A}_{TB}={A}_{B}+{A}_{T}\phantom{\rule{0ex}{0ex}}{A}_{TB}={A}_{B}+{A}_{B}\phantom{\rule{0ex}{0ex}}{A}_{TB}=2{A}_{B}\phantom{\rule{0ex}{0ex}}$

So, the area of the base and top is twice the base area.

Now we still have n rectangular sides. This means we have to calculate the area of each rectangle. This would even be more stressful as the number of sides increases.

$Areaofface1=Side1\times height\phantom{\rule{0ex}{0ex}}Areaofface2=Side2\times height\phantom{\rule{0ex}{0ex}}Areaofface3=Side3\times height\phantom{\rule{0ex}{0ex}}Areaofface4=Side4\times height\phantom{\rule{0ex}{0ex}}.\phantom{\rule{0ex}{0ex}}.\phantom{\rule{0ex}{0ex}}.\phantom{\rule{0ex}{0ex}}Areaofface\mathit{n}=Side\mathit{n}\times height\phantom{\rule{0ex}{0ex}}$

Do you like stress? Well, I don't .

So to cut the labor down, something is constant. The height is constant, since we are going to sum all areas why not find the sum of all the sides and multiply by the height. This means that

id="2899374" role="math" $Totalrec\mathrm{tan}gularbodyareaofaprism=(Side1\times height)+(Side2\times height)+(Side3\times height)..+Siden\times height)\phantom{\rule{0ex}{0ex}}Totalrec\mathrm{tan}gularbodyareaofaprism=height(Side1+Side2+Side3+Side4...+Siden\left)\phantom{\rule{0ex}{0ex}}\right(Side1+Side2+Side3+Side4...+Siden)=Perimeterofbasesurface\phantom{\rule{0ex}{0ex}}Totalrec\mathrm{tan}gularbodyareaofaprism=height(Perimeterofbasesurface)\phantom{\rule{0ex}{0ex}}$

Where h is the height of a prism, A_{B} is the base area, and P_{B} is the perimeter of the prism base, the total surface area of a prism is

${A}_{P}=2{A}_{B}+{P}_{B}h$

An illustration of the height and base of a prism for determining the surface area, StudySmarter Originals

## What is the surface area of a triangular prism?

If h is the height of a prism, A_{B} is the base area, and P_{B} is the perimeter of the prism base, the total surface area of a prism can be calculated using the following formula:

${A}_{P}=2{A}_{B}+{P}_{B}h$

But we have to customize this formula to suit a triangle since a triangular prism has the base of a triangle. Since the area of a triangle A_{t} with a base b and height h_{t} is

${A}_{t}=\frac{1}{2}b\times {h}_{t}$

and the perimeter of a triangle P_{t} with a, b, c is

${P}_{t}=a+b+c$

then the total surface area of a triangular prism A_{Pt }would be

${A}_{Pt}=2(\frac{1}{2}b\times {h}_{t})+h(a+b+c)\phantom{\rule{0ex}{0ex}}{A}_{Pt}=\overline{)2}(\frac{1}{\overline{)2}}b\times {h}_{t})+h(a+b+c)\phantom{\rule{0ex}{0ex}}{A}_{Pt}=(b\times {h}_{t})+h(a+b+c)$

Note that h_{t} is the height of the triangular base while h is the height of the prism itself.

An illustration of the area of a triangular prism, StudySmarter Originals

The total surface area of a triangular prism is:

**sum of (product of base and height of triangular base) and (product of height of prism and perimeter of triangle)**

Find the total surface area of the figure below.

**Solution:**

The total surface area of a triangular prism A_{Pt }is

${A}_{Pt}=(b\times {h}_{t})+h(a+b+c)$

b is 6 m,

h_{t} is 4 m,

h is 3 m,

a is 5 m,

and c is also 5 m (Isosceles triangular base)

Then substitute into your formula and solve.

${A}_{Pt}=(6m\times 4m)+3m(5m+6m+5m)\phantom{\rule{0ex}{0ex}}{A}_{Pt}=(24{m}^{2})+3m(16m)\phantom{\rule{0ex}{0ex}}{A}_{Pt}=24{m}^{2}+48{m}^{2}\phantom{\rule{0ex}{0ex}}{A}_{Pt}=\mathbf{72}\mathbf{}{\mathit{m}}^{\mathbf{2}}$

## What is the surface area of a rectangular prism?

A rectangular prism is called a **cuboid** if it has a rectangular base or a **cube **if it has a square base with the height of the prism equal to the side of the square base.

Where h is the height of a prism, A_{B} is the base area, and P_{B} is the perimeter of the prism base, the total surface area of a prism can be calculated using the following formula:

${A}_{P}=2{A}_{B}+{P}_{B}h$

But we have to customize this formula to suit a rectangle since a rectangular prism has the base of a rectangle. Since the area of a rectangle A_{r} with a base b and height h_{r} is

${A}_{r}=b\times {h}_{r}$

and the perimeter of the same rectangle P_{r} is

${P}_{r}=2(b+{h}_{r})$

then the total surface area of a triangular prism A_{Pr }would be

${A}_{Pr}=2(b\times {h}_{r})+h\left(2\right(b+{h}_{r}\left)\right)\phantom{\rule{0ex}{0ex}}{A}_{Pr}=2(b\times {h}_{r})+2h(b+{h}_{r})\phantom{\rule{0ex}{0ex}}{A}_{Pr}=2\left(\right(b\times {h}_{r})+h(b+{h}_{r}\left)\right)$

Note that h_{r} is the height of the rectangular base while h is the height of the prism itself. Also, the base b and the height h_{r} of the rectangular base is otherwise known as the **breadth** and **length** of the rectangular base.

The total surface area of a rectangular prism is:

**Twice the sum between the product of the base and the height of the rectangular base and the product of the height of the prism and the sum of the base and the height of the rectangular base**

Find the total surface area of the figure below.

**Solution:**

The total surface area of a rectangular prism A_{Pr} is

${A}_{Pr}=2\left(\right(b\times {h}_{r})+h(b+{h}_{r}\left)\right)$

b is 10 cm,

h_{r} is 6 cm,

and h is 8 cm

Then substitute into your formula and solve.

id="2899393" role="math" ${A}_{Pr}=2\left(\right(10cm\times 6cm)+8cm(10cm+6cm\left)\right)\phantom{\rule{0ex}{0ex}}{A}_{Pr}=2\left(\right(60c{m}^{2})+8cm(16cm\left)\right)\phantom{\rule{0ex}{0ex}}{A}_{Pr}=2(60c{m}^{2}+128c{m}^{2})\phantom{\rule{0ex}{0ex}}{A}_{Pr}=\mathbf{376}\mathbf{}\mathit{c}{\mathit{m}}^{\mathbf{2}}$

Note, for other types of shapes, just input their respective areas and find their perimeters and apply the general formula

${A}_{P}=2{A}_{B}+{P}_{B}h$

you would surely arrive at the right answer.

## Examples of surface area of prisms

You are advised to try as many examples as possible to increase your competence in solving problems on the surface area of prisms. Below are some examples to help you out.

Find the total surface area of the figure below.

**Solution:**

This is a triangular prism. Before we can go ahead to calculate its total surface area we need to find the sides of its triangular base.

Since the height is 9 cm and it is an isosceles triangle, we can use Pythagoras theorem to find the rest of the sides. Let x be the unknown side.

then x is

${x}^{2}={5}^{2}+{9}^{2}\phantom{\rule{0ex}{0ex}}x=\sqrt{{5}^{2}+{9}^{2}}\phantom{\rule{0ex}{0ex}}x=\sqrt{25+81}\phantom{\rule{0ex}{0ex}}x=\sqrt{106}\phantom{\rule{0ex}{0ex}}x=10.3$

Now we know the other side we can apply our formula

${A}_{Pt}=(b\times {h}_{t})+h(a+b+c)$

b is 10 cm,

h_{t} is 9 cm,

h is 6 cm,

a is 10.3 cm,

and c is also 10.3 cm (Isosceles triangular base)

Now substitute into the formula and solve.

${A}_{Pt}=(10cm\times 9cm)+6cm(10.3cm+10cm+10.3cm)\phantom{\rule{0ex}{0ex}}{A}_{Pt}=(90c{m}^{2})+6cm(30.6cm)\phantom{\rule{0ex}{0ex}}{A}_{Pt}=90c{m}^{2}+183.6c{m}^{2}\phantom{\rule{0ex}{0ex}}{A}_{Pt}=\mathbf{273}\mathbf{.}\mathbf{6}\mathbf{}\mathit{c}{\mathit{m}}^{\mathbf{2}}$

Find the length of a cube if its total surface area is 150 cm^{2}.

**Solution:**

Remember that a type of rectangular prism which has all its sides equal. Knowing that the total surface area of a rectangular prism A_{Pr} is

${A}_{Pr}=2\left(\right(b\times {h}_{r})+h(b+{h}_{r}\left)\right)$

then for a cube which has all its sides equal,

$b={h}_{r}=h$

So,

${A}_{Pr}=2\left(\right(b\times b)+b(b+b\left)\right)\phantom{\rule{0ex}{0ex}}{A}_{Pr}=2({b}^{2}+b(2b\left)\right)\phantom{\rule{0ex}{0ex}}{A}_{Pr}=2({b}^{2}+2{b}^{2})\phantom{\rule{0ex}{0ex}}{A}_{Pr}=2\left(3{b}^{2}\right)\phantom{\rule{0ex}{0ex}}{A}_{Pr}=6{b}^{2}$

We are told that the total surface area A_{Pr} is 150 cm^{2} so each side would be

${A}_{Pr}=6{b}^{2}\phantom{\rule{0ex}{0ex}}150c{m}^{2}=6{b}^{2}\phantom{\rule{0ex}{0ex}}\frac{150c{m}^{2}}{6}=\frac{6{b}^{2}}{6}\phantom{\rule{0ex}{0ex}}{b}^{2}=25c{m}^{2}\phantom{\rule{0ex}{0ex}}b=\sqrt{25c{m}^{2}}\phantom{\rule{0ex}{0ex}}b=\mathbf{5}\mathbf{}\mathit{c}\mathit{m}$

This means that the cube which has a total surface area as 150 cm^{2} has a length of **5 cm**.

## Surface of Prisms - Key takeaways

- A prism is a 3-dimensional geometrical figure that has a
**constant cross-section**throughout itself. A prism has identical ends and**flat faces**. - The surface area of any prism can be calculated with the formula $surfacearea=(basearea\times 2)+\left(baseperimter\times length\right)$

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##### Frequently Asked Questions about Surface Area of Prism

What is the formula for finding the surface area of prism?

Surface area= (base area x 2)+(base perimeter x length)

How to calculate the surface area of triangular prism?

For this, you will need to find the base area by calculating 1/2 x b x h and the base perimeter by adding all the sides of the base triangle. Then you can use the formula surface area= (base area x 2)+(base perimeter x height)

What are the properties of a prism?

A prism has a constant cross-section and flat surfaces.

What is an example of the surface area of a prism?

An example of the surface area of a prism is using a cube of 3 cm. A cube has 6 square faces and the area of each square would be the product of 3 and 3 which gives 9 cm^{2}. Since you have six sides then the total surface area is the product of 6 and 9 cm^{2} which gives 54 cm^{2}.

What is the surface area of a prism?

The area of surfaces of prisms is the total plane surface occupied by the sides of 3-dimensional geometrical figures that have **constant cross-sections **throughout their body.

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