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In these lessons, we will learn how to

- calculate the surface area of prisms: rectangular prisms, triangular prisms, trapezoidal prisms, hexagonal prisms etc.
- solve problems about prisms.
- calculate the surface area of prisms using nets.

A prism is a solid that has two parallel faces which are congruent polygons at both ends. These faces form the bases of the prism. A prism is named after the shape of its base. The other faces are in the shape of parallelograms. They are called lateral faces.

The following diagrams show a triangular prism and a rectangular prism.

A right prism is a prism that has its bases perpendicular to its lateral surfaces.

When we cut a prism parallel to the base, we get a cross section of a prism. The cross section has the same size and shape as the base, as can be seen in the following diagrams.

The surface area of a solid is the total area of all its external faces.

To calculate the surface area of a prism we need to:

**Step 1 **:
Determine the shape of each face.

**Step 2** : Calculate
the area of each face.

**Step 3** : Add up all
the areas to get the total surface area.

We can also use the formula

Surface area of prism = 2 × area of base + perimeter of base × height

Worksheet to calculate the surface area and volume of a rectangular prism

**Example: **

Calculate the surface area of the following prism.

**Solution: **

There are 2 triangles with the base = 4 cm and height = 3 cm.

Area of the 2 bases

= 12 cm^{2}

1 rectangle with length = 7 cm and width = 5 cm

Area = *lw* = 7 × 5 = 35 cm^{2}

1 rectangle with length = 7 cm and width 3 m

Area = *lw* = 7 × 3 = 21 cm^{2}

1 rectangle with length = 7 cm and width 4 m

Area = *lw* = 7 × 4 = 28 cm^{2}

The total surface area is 12 + 35 + 21 + 28 = 96 cm^{2}

We can also use the formula

Surface area of prism = 2 × area of base + perimeter of base × height

= 2 × 6 + (3 + 4 + 5) × 7 = 96 cm^{2}

**Example: **

The diagram shows a prism whose base is a trapezoid. The surface
area of the prism is 72 cm 2. Find the value of *x*.

*
*

**Solution: **

There are 2 rectangles with length = 5 cm and width = 3 cm

Area = 2 × 5 × 3 = 30 cm^{2}

There is one rectangle with length = 5 cm and width = 4 cm

Area = 5 × 4 = 20 cm^{2}

There is one rectangle with length = 5 cm and width = 2 cm

Area = 5 × 2 = 10 cm^{2}

There are two trapezoids.

Area = cm^{2} = 6*x* cm^{2}

Sum of area

30 + 20 + 10 + 6*x* = 72

60 + 6*x* = 72

*x* = 2

The value of *x* is 2.

This video shows how to find the surface area of prisms: cuboid (or rectangular prism), triangular prism, trapezoidal prism.

The following video shows how to find the surface area of a rectangular prism.

This video shows how to find the surface area of a triangular prism.

This video shows how to find the surface area of a triangular prism using the formula SA = ab+(s1+s2+s3)h.

This video shows how to find the surface area of a pentagonal prism.

This video shows how to find the surface area of a hexagonal prism.

This video shows how to find the surface area of a octagonal prism.

Problem: Bria's camping tent is not waterproof, so she wants to put tarp over it, in case it rains. The trap will cover two of the faces. Which size trap should she buy?

Problem: Use the given dimensions to determine the surface area and volume of the candy box.

This video shows how to find the surface area of a cube, rectangular prism and triangular prism using nets.

This video shows how to find the surface area of a pentagonal prism using a net.

This video shows how to find the surface area of a hexagonal prism using a net.