This section is all about the volume and surface area of prisms, cylinders, pyramids, cones and spheres! All of these shapes has their own unique properties and this section is about understanding and calculating the volumes and surface areas.

Key Concepts

In this unit you should learn to…

work out the volume and surface area of 3 shapes including prisms, cylinders, pyramids, cones and spheres

Use the above to solve problems in 3 dimensions

Essentials

Slides Gallery - Part 1 - Prisms

1. What are Prisms

This is a video explaining the properties of a prism and looking at different kinds of prism.

2. Volume of Prisms

Here we learn about how to use the properties of prisms to calculate their volumes. the video goes through some examples.

3.Surface area of prisms

Here we learn about calculating surface area by examining the nets of different kinds of prisms.

Volume and Surface area of Cylinders

Here are some slides and videos going over cylinders. Cylinders have the same properties as Prisms, but because the cross section is a circle which is not a straight edge there is a bit of extra thinking to do.

4. Volume of Cylinders

Here we look at applying the same principles as with prisms to cylinders with their circular base.

5. Surface area of cylinders

The nets of cylinders just take a little bit more thought.....

Pyramids and Cones

This section of the page moves on to pyramids and their properties. We will look at calculating their respective volumes and surface areas.

6. What are pyramids?

Defining the key properties of these shapes and looking at a few different ones.

7. Volume of a pyramid and cone

This is calculated by taking a third of the volume of the prism it would fit in. This video shows this in action.

8. Surface area of a Pyramid

Here we look at the nets of different pyramids so we can see how surface area can be calculated.

Cones and Spheres

In this last section we deal with these shapes with curved surface areas, their properties, volume and surface area

9. Surface Area of a Cone

Unpacking the net of a cone is lovely example of mathematical links. In this video, we explore the links between the variables and show how to calculate this.

10. Volume and Surface area of a Sphere

The sphere is in one way the hardest to deal with, but in another can be done with some startlingly simple formulae!

Summary

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Test Yourself

Self Checking quiz

1

a) What is the volume of the cuboid in the diagram below?

b) What is its surface area?

Write a number only

a) Volume = cm^{3}

b) Surface area = cm^{2}

The volume of the base is 5 x 2, then multiply by the height, 6, giveives volume = 5 x 2 x 6

b) 2 faces are 5 x 2, 2 faces are 6 x 2 and 2 faces are 5 x 6, giving a total of 20 + 24 + 60 = 104cm^{2}

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2

Drag the correct name of the shape in to the right box.

Shape A

Triangular prismPyramid

Shape B

CylinderCone

Shape C

CuboidOctangonal based pyramid

Shape D

Rectangular based PyramidSphere

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3

The following right angled triangle is one face of a triangular prism whose length is 12 cm....

a) Calculate the length of the hypotenuse of the triangle

b) What is the area of the triangular cross section?

c) What is the volume of the prism?

d) What is the total surface area of the prism?

Write numbers only

a) Length of hypotenuse is cm

b) The area of the triangular cross section is cm^{2}

c) The volume of the prism is cm^{3}

d)The total surface area of the prism is cm^{2}

a) Using pythagoras theorem, h^{2} = 3^{2} + 4^{2}

b) Area = (3 x 4)/2 = 6

c) Volume is corss sectional area x length, so V = 6 x 12 = 72

d) There are 5 faces, 2 triangles on each end each 6, the bottom which is 3 x 12, the back which is 4 x 12 and the sloping face which is 5 x 12, so 6 + 6 + 36 + 48 + 60 = 156

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4

A triangular prism has a volume of 300 cm^{3} and a length of 15cm.

a) What is the area of the triangular cross section?

b) If the triangular corss section has a height of 5cm, how long is its base?

a) Area of triangular face is cm^{2}

b) the base of the triangular criss section is cm

a) Volume = Cross section x length, so 300 = Cross section x 15. Cross section = 300/15

b) Area of truiangle is base x height / 2. If base x 5/2 = 20 then base must be 8

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5

Consider the following cylinder with a radius of 7 cm and a heigth of 15 cm

a) Work out the volume of the cylinder

b) Work out the surface are of the cylinder

Give your answers to 3sf

a) Volume = cm^{3}

b) Surface area = cm^{2}

a) Volume of a cylinder = (pi)r^{2}h. in this case pi x 7^{2 }x 15 = 2039.07060 which is 2040 to 3sf

b) Surface area of a cylinder = 2 (pi)r^{2} + 2(pi)rh, substitute r = 7 and h = 15 and you get SA = 4926.01728 which rounds to 4930 to 3sf

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6

Consider the following rectangular based Pyramid and its net. CD = 4cm, BC = 2cm, EF = 3cm

a) Work out the volume of the pyramid

b) What is the perpendicualr height, h, of triangle ECD? (give answer to 3sf)

c) What is the perpendicualr height, p, of triangle EBC? (give answer to 3sf)

d) Calculate the surface area of the Pyramid using rounded answers to the previous questions

A sphere and a cone have the same radius, 7cm, and the same volume. What is the height of the cone? Do not round until your final answer then give your answer to 3sf.