The volume of any prism or cylinder can be calculated using the formula, by simply substituting the area of the base for B:

\displaystyle V = Bh

\bm{B}

area of the base

\bm{h}

perpendicular height between bases

Here are some specific volume formulas:

Use the applet below to explore the volume of prisms with triangular and rectangular bases.

- Set the dimensions of the triangular prism: b=5,\, h=8,\,\text{Height}=5. What is the volume?
- Set the dimensions of the rectangular prism: \text{width}=2,\,\text{length}=10,\,\text{Height}=5. What do you notice about the Height and the volume of the triangular prism and rectangular prism from part (1)?
- Find a set of dimensions for the two prisms, so that they have the same volume, but different heights.
- If you double the Height of both shapes, what do you notice about the volumes of the triangular prism and the rectangular prism?

Find the volume of a cylinder rounded to one decimal place if its radius is 5\text{ cm} and its height is 13\text{ cm.}

Worked Solution

A 13 \text{ cm} concrete cylindrical pipe has an outer radius of 6 \text{ cm} and an inner radius of 4 \text{ cm} as shown. Find the volume of concrete required to make the pipe, rounded to two decimal places.

Worked Solution

Find the volume of the triangular prism shown.

Worked Solution

Find the volume of this figure, rounded to two decimal places.

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A prism has a volume of 990 \text{ cm}^3. If it has a base area of 110 \text{ cm}^2, find the height of the prism.

Worked Solution

A manufacturer is designing a new storage box to replace its current cube-shaped model.

The volume of the new storage box will be 27 times greater than that of the original.

The new storage box will have the same length and width as the original box.

What will be the height of the new storage box compared to the original box?

Worked Solution

Idea summary

The volume, V, of a prism or cylinder is calculated using the formula V=Bh, where B represents the area of the base and h represents the height of the prism.

Drag the sliders and check the boxes to explore the applet.

Create a triangular prism by changing N to 3 and dragging the sliders to close the net. How can you use the net to find the surface area of the prism?

What happens as you drag the N slider to its largest value?

A **net** is a diagram of the faces of a three-dimensional figure arranged in such a way that the diagram can be folded to form the three-dimensional figure. We can find the surface area from a net by calculating the sum of the areas of the **lateral faces** and the bases.

A cube and its net

A right cylinder and its net

Note that even some solids with curved faces, such as cylinders, have nets consisting of flat, two-dimensional shapes. Additionally, a solid can be represented with multiple different, equivalent nets.

The surface area of a cylinder can be calculated identically; by adding the area of two circular bases to the product of the circumference and the perpendicular height between bases.

We can find the **lateral area** of a prism or a cylinder by ignoring the bases.

We can write a general formula for the surface area of prisms and cylinders:

\displaystyle SA=LA+2B

\bm{LA}

total lateral area

\bm{B}

area of one base

Consider the can of tuna shown below:

a

Draw the net of the can of tuna and label its dimensions.

Worked Solution

b

Find the area of each part of the net and find the total surface area of tin that a company must produce per can of tuna.

Worked Solution

c

The formula for finding the surface area of a prism or cylinder is SA= 2B + Ph, where SA represents surface area, B represents the area of the base, P represents the perimeter of the base, and h represents the height of the prism or cylinder. Explain what each part of the formula for surface area represents, and relate it to finding the surface area for the can of tuna.

Worked Solution

Find the surface area of the triangular prism shown.

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Yuki is decorating a wedding cake with some icing. He wants to cover the outer facing surface of the cake with a layer of icing that is an eighth of an inch thick.

Determine how much icing Yuki will need to decorate the cake.

Worked Solution

A rectangular prism has a length 16 \text{ cm}, a width of 10\text{ cm}, and a height of 9\text{ cm}. If the height of the rectangular prism is increased by 4\text{ cm} while the other dimensions remain unchanged, calculate and compare the new volume and surface area to the original.

a

Compare the new volume of the rectangular prism to the original.

Worked Solution

b

Compare the new surface area of the rectangular prism to the original.

Worked Solution

Idea summary

We can calculate the surface area of any prism or cylinder using the formula

\displaystyle SA=LA+2B

\bm{LA}

total lateral area

\bm{B}

area of one base