Volume & Surface Area (DP IB Maths: AA HL)

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Amber

Last updated

30 March 2023

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Volume of 3D Shapes

What is volume?

The volume of a 3D shape is a measure of how much 3D space it takes up
- A 3D shape is also called a solid
You need to be able to calculate the volume of a number of common shapes

How do I find the volume of cuboids, prisms and cylinders?

A prism is a 3-D shape that has two identical base shapes connected by parallel edges
- A prism has the same base shape all the way through
- A prism takes its name from its base
To find the volume of any prism use the formula:

Volume of a prism = Ah

- Where A is the area of the cross section and h is the base height
  - h could also be the length of the prism, depending on how it is oriented
- This is in the formula booklet in the prior learning section at the beginning
- The base could be any shape so as long as you know its area and length you can calculate the volume of any prism

Prism volume

Note two special cases:
- To find the volume of a cuboid use the formula:

$\begin{array}{rcl} Volume of a cuboid & = & length \times width \times height \\ V & = & l w h \end{array}$

- The volume of a cylinder can be found in the same way as a prism using the formula:

$Volume of a cylinder = π r^{2} h$

- where $r$ is the radius, $h$ is the height (or length, depending on the orientation
- Note that a cylinder is technically not a prism as its base is not a polygon, however the method for finding its volume is the same
Both of these are in the formula booklet in the prior learning section

How do I find the volume of pyramids and cones?

In a right-pyramid the apex (the joining point of the triangular faces) is vertically above the centre of the base
- The base can be any shape but is usually a square, rectangle or triangle
To calculate the volume of a right-pyramid use the formula

$V = \frac{1}{3} A h$

- Where A is the area of the base, h is the height
- Note that the height must be vertical to the base

A right cone is a circular-based pyramid with the vertical height joining the apex to the centre of the circular base
To calculate the volume of a right-cone use the formula

$V = \frac{1}{3} π r^{2} h$

- Where $r$ is the radius, $h$ is the height

These formulae are both given in the formula booklet

How do I find the volume of a sphere?

To calculate the volume of a sphere use the formula

$V = \frac{4}{3} π r^{3}$

- Where r is the radius
  - the line segment from the centre of the sphere to the surface
- This formula is given in the formula booklet

Examiner Tip

Remember to make use of the formula booklet in the exam as all the volume formulae you need will be here
- Formulae for basic 3D objects (cuboid, cylinder and prism) are in the prior learning section
- Formulae for other 3D objects (pyramid, cone and sphere) are in the Topic 3: Geometry section

Worked example

A dessert can be modelled as a right-cone of radius 3 cm and height 12 cm and a scoop of ice-cream in the shape of a sphere of radius 3 cm. Find the total volume of the ice-cream and cone.

diagram-for-we-3-2-2

3-2-2-ai-sl-volume-we-solution

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Surface Area of 3D Shapes

What is surface area?

The surface area of a 3D shape is the sum of the areas of all the faces that make up a shape
- A face is one of the flat or curved surfaces that make up a 3D shape
- It often helps to consider a 3D shape in the form of its 2D net

How do I find the surface area of cuboids, pyramids and prisms?

Any prisms and pyramids that have polygons as their bases have only flat faces
- The surface area is simply found by adding up the areas of these flat faces
- Drawing a 2D net will help to see which faces the 3D shape is made up of

How do I find the surface area of cylinders, cones and spheres?

Cones, cylinders and spheres all have curved faces so it is not always as easy to see their shape
- The net of a cylinder is made up of two identical circles and a rectangle
- The rectangle is the curved surface area and is harder to identify
- The length of the rectangle is the same as the circumference of the circle
- The area of the curved surface area is

$A = 2 π r h$

- - where r is the radius, h is the height
- This is given in the formula book in the prior learning section
- The area of the total surface area of a cylinder is

$A = 2 π r h + 2 π r^{2}$

- This is not given in the formula book, however it is easy to put together as both the area of a circle and the area of the curved surface area are given

The net of a cone consists of the circular base along with the curved surface area
- The area of the curved surface area is

$A = π r l$

- - Where r is the radius and l is the slant height
- This is given in the formula book
  - Be careful not to confuse the slant height, l, with the vertical height, h
  - Note that r, h and l will create a right-triangle with l as the hypotenuse
- The area of the total surface area of a cone is

$A = π r l + π r^{2}$

- This is not given in the formula book, however it is easy to put together as both the area of a circle and the area of the curved surface area are given

To find the surface area of a sphere use the formula

$A = 4 π r^{2}$

- - where r is the radius (line segment from the centre to the surface)
- This is given in the formula booklet, you do not have to remember it

Examiner Tip

Remember to make use of the formula booklet in the exam as all the area formulae you need will be here
- Formulae for basic 2D shapes (parallelogram, triangle, trapezoid, circle, curved surface of a cylinder) are in the prior learning section
- Formulae for other 2D shapes (curved surface area of a cone and surface area of a sphere ) are in the Topic 3: Geometry section

Worked example

In the diagram below ABCD is the square base of a right pyramid with vertex V . The centre of the base is M. The sides of the square base are 3.6 cm and the vertical height is 8.2 cm.

sa-diagram-for-we-3-2-2