11. Solids

Lesson

Volume is the amount of space taken up by a $3$3D object. While there are many specific formulas for particular prisms, in general, we can always come back to the formula below.

Volume of a prism

$\text{Volume of any prism }$Volume of any prism | $=$= | $\text{Area of Base }\times\text{Height }$Area of Base ×Height |

$V$V |
$=$= | $A_{base}\times h$Abase×h |

Find the volume of the cube shown.

Find the volume of the prism by finding the base area first.

A cylinder is very similar to a prism (except for the lateral face), so the volume can be found using the same concept we have already learned.

Volume of a cylinder

$\text{Volume of Cylinder }$Volume of Cylinder | $=$= | $\text{Area of Base }\times\text{Height of Prism }$Area of Base ×Height of Prism |

$\text{Volume of Cylinder }$Volume of Cylinder | $=$= | $\pi r^2\times h$πr2×h |

$\text{Volume of Cylinder }$Volume of Cylinder | $=$= | $\pi r^2h$πr2h |

You are at the local hardware store to buy a can of paint. After settling on one product, the salesman offers to sell you a can that is either double the height or double the radius (your choice) of the one you had decided on for double the price. Assuming all cans of paint are filled to the brim, is it worth taking up his offer?

**Think:** If so, would you get more paint for each dollar if you chose the can that was double the radius or the can that was double the height?

**Do:**

**Doubling the height: $h\to2h$ h→2h**

Since the volume of a cylinder is given by the formula $\pi r^2h$π`r`2`h`, if the height doubles, the volume becomes $\pi r^2\times2h=2\pi r^2h$π`r`2×2`h`=2π`r`2`h`

If we double the height, we double the volume.

**Doubling the radius:** $r\to2r$`r`→2`r`

Whereas if the radius doubles, the volume becomes $\pi\left(2r\right)^2h=4\pi r^2h$π(2`r`)2`h`=4π`r`2`h`.

If we double the height, we quadruple the volume.

**Reflect**: We should take the offer for and double the radius of the paint can as we get four times the paint for double the cost.

To see how changes in height and radius affect the volume of a can to different extents, try the following interactive. You can vary the height and radius by moving the sliders around.

Find the volume of a cylinder correct to 1 decimal place if its diameter is $2$2 cm and its height is $19$19 cm.

Find the volume of a cylinder with radius $7$7 cm and height $15$15 cm, correct to 2 decimal places.

Calculate the volume of the solid correct to two decimal places.

It is probably worthwhile to remind ourselves of the units that are often used for calculations involving volume.

Units for Volume

**cubic millimeters = mm ^{3}**

(picture a cube with side lengths of 1 mm each - pretty small this one!)

**cubic centimeters = cm ^{3}**

(picture a cube with side lengths of 1 cm each - about the size of a dice)

**cubic meters = m ^{3} **

(picture a cube with side lengths of 1 m each - what could be this big?)

AND to convert to **capacity - 1cm ^{3} = 1mL**

Find the volume of the figure shown.

A hole is drilled through a rectangular box forming the solid shown. Find the volume of the solid correct to 2 decimal places.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★