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VCE 11 General 2023

10.07 Volume

Lesson

Prisms

A prism is defined as a solid geometric figure whose two end faces are exactly the same shape, straight and parallel.

 

Volume of any prism

If the solid is a prism, its volume can be worked out by using the following rule.

Volume of prism

$\text{Volume }=\text{Area of base }\times\text{Height }$Volume =Area of base ×Height

where the base is one of the two faces that are identical and parallel to one another and the height is perpendicular to the base.

In this shape, the base is the L-shaped section. It is possible to calculate the volume of an irregular prism by first calculating the area of the base and multiplying this by the height.

$\text{Volume of an irregular prism }=\text{Area of the base (L shape) }\times\text{Height of the prism }$Volume of an irregular prism =Area of the base (L shape) ×Height of the prism

Volume of common prisms

$\text{Volume of rectangular prism }=\text{Length }\times\text{Width }\times\text{Height }$Volume of rectangular prism =Length ×Width ×Height

$\text{Volume of triangular prism }=\text{Area of the triangle }\times\text{Height of the prism }$Volume of triangular prism =Area of the triangle ×Height of the prism

$\text{Volume of trapezium based prism }=\text{Area of the trapezium }\times\text{Height of the prism}$Volume of trapezium based prism =Area of the trapezium ×Height of the prism

Use the following applet to experiment with how the volume of a triangular prism is affected by its base and height. You can vary the dimensions by moving the sliders and see the volume calculations by checking the checkbox.

 

Practice questions

Question 1

Find the volume of the cube shown.

A three-dimensional cube with edges depicted in a green outline. The front bottom edge of the cube is labeled with the measurement of $12$12 cm.

Question 2

Find the volume of the triangular prism shown.

A right-angled triangular prism with height of 2 cm, base of 4 cm and a length of 8 cm.

Question 3

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

A three-dimensional trapezoid prism is depicted. The trapezoid is facing front. The trapezoid has a bottom base measuring 16 cm, and a top base measuring 13 cm. The height of the trapezoid is measured 5 cm. The depth of the shape is measured 3 cm.

 

Cylinders

A cylinder is a type of prism where the base shape is a circle.

Volume of cylinder

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

To see how changes in height and radius affect the volume of a cylinder, try the following interactive. You can vary the height and radius by moving the sliders and view the volume calculations.

 

Practice questions

Question 4

Find the volume of the cylinder shown.

A cylinder with two dimensions labeled. The radius of the cylinder is measuring $3$3 cm, and the height of the cylinder is measuring $13$13 cm.
  1. Round your answer to two decimal places.

Question 5

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

A three-dimensional representation of a concrete cylindrical pipe. The pipe has a length of $13$13 cm, with the front end visibly showing a hollow interior. The inner radius of the pipe is labeled as $4$4 cm, while the outer radius is labeled as $6$6 cm. 

Exploration

You are about to go camping for the first time and need to buy a tent. You visit the local store that sells camping gear and one tent immediately attracts your attention but it appears to be a little too small for your liking. When you ask one of the salesmen for a tent with a larger volume he points you to a tent that has a similar length but double the height and that he claims is the largest tent in store. When you ask another salesman he points you to a tent that has a similar height to your original choice but double the length and that he claims is the largest tent in store. Who is right? Can they both be right?

If you think back to the formula for the volume of a triangular prism, $\frac{1}{2}\times\text{base }\times\text{height }\times\text{length of prism }$12×base ×height ×length of prism if the prism is laid out like a tent, you will notice that when the height doubles, so too will the volume; and when the length doubles, so too will the volume. So in the case of the salesmen, both of them could be right because both of the tents they point out have the same volume.

 

Volume of pyramids

A pyramid is formed when the vertices of a polygon are projected up to a common point (called a vertex). A right pyramid is formed when the apex is perpendicular to the midpoint of the base.

 

Volume of pyramid

$\text{Volume of Pyramid }=\frac{1}{3}\times\text{Area of base }\times\text{Height }$Volume of Pyramid =13×Area of base ×Height

 

Practice questions

Question 6

Find the volume of the square pyramid shown.

A triangular pyramid is depicted with its faces visible and outlined. A vertical dashed line, representing the height, is drawn from the apex of the pyramid perpendicular to the base and is labeled as 6$cm$cm. One side of the base is labeled as 10$cm$cm. All the sides of the base are drawn with a single tick mark indicating that the measurements of the sides are all equal. 

Question 7

A small square pyramid of height $4$4 cm was removed from the top of a large square pyramid of height $8$8 cm forming the solid shown. Find the exact volume of the solid.

A  square pyramid is depicted. The top portion of the pyramid is removed, indicated by dashed lines, creating a new top base, with its side measuring 4 cm. The bottom base of the square pyramid has a side measuring 8 cm. The vertical height of the pyramid is composed of two measurements, the removed top smaller pyramid measuring 4 cm, and the bottom part measuring 4 cm. 
  1. Give your answer in exact form.

 

Volumes of spheres

The volume of a sphere with radius $r$r can be calculated using the following formula.

Volume of sphere

$\text{Volume of sphere }=\frac{4}{3}\pi r^3$Volume of sphere =43πr3

Practice questions

Question 8

Find the volume of the sphere shown.

Round your answer to two decimal places.

A sphere is shown. The radius measures 3 cm.

Question 9

A sphere has a radius $r$r cm long and a volume of $\frac{343\pi}{3}$343π3 cm3. Find the radius of the sphere.

Round your answer to two decimal places.

Enter each line of working as an equation.

Outcomes

U2.AoS4.6

formulas for the volumes and surface areas of solids (spheres, cylinders, pyramids, prisms) and their application to composite objects

U2.AoS4.13

calculate the perimeter, areas, volumes and surface areas of solids (spheres, cylinders, pyramids and prisms and composite objects) in practical situations, including simple uses of Pythagoras’ in three dimensions

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