Year 12 Further Maths Notes
Chapter 7 – Geometry
Properties of Triangles and Polygons
For a regular polygon, with 𝑛 sides, the size of the interior angle (𝜃) can be found using the following equation
360
𝜃 = 180 −
𝑛
The exterior angle (∅) can be found using
360
∅=
𝑛
Definitions of common terms
Common Notations and Rules
Exercise 7A - 1, 2, 3, 4, 5
Area and Perimeter
Perimeter
– a measure of the distance around the outside of an enclosed shape.
For a square, the perimeter = 4𝑙
For a rectangle, the perimeter = 2𝑙 + 2𝑤
For a circle, the perimeter is called the circumference and can be found using the equation, circumference
= 2𝜋𝑟
Area
– a measure of the amount of space contained within an enclosed shape.
Composite Shapes
A composite shape is one that is made up of more than one regular shape joined together.
Conversion of units of area
If you are unsure of how to convert from square metres to square centimetres then use the following diagram
to help out.
Exercise 7B – 1, 2, 3, 5, 10
Total Surface Area
To find the total surface area of a shape not given in the ones above, try drawing the net of the object and
then find the area of the net. This area will be representative of the total surface area.
Exercise 7C – 1, 2, 3, 5, 8
Volume
To convert between units of volume, use the following diagram to assist.
Prism
Pyramid
Sphere
To find the volume of a prism we use the following rule
𝑉 =𝐴×𝐻
Where 𝐴 is the cross-sectional area of the prism and 𝐻 is the height of the prism.
To find the volume of a pyramid we use the following rule
1
𝑉 = ×𝐴×𝐻
3
Where 𝐴 is the area of the pyramid base and 𝐻 is the altitude of height of the pyramid.
To find the volume of a sphere we use the following rule
4
𝑉 = 𝜋𝑟 3
3
Where 𝑟 is the radius of the sphere.
Exercise 7D – 1, 2a), d), e), 3a), c), 4d), 5a), c), e), 6a), b), e), g), 9
Similar Figures
Similar figures are objects that have the same shape but different size.
For two figures to be similar they must satisfy the following 2 properties
The ratio of the side lengths must be equal
𝐴′𝐵′ 𝐵′𝐶′ 𝐶′𝐷′ 𝐷′𝐴′
=
=
=
𝐵𝐶
𝐴𝐵
𝐶𝐷
𝐷𝐴
The corresponding angles must be equal
< 𝐴 =< 𝐴′ ,
< 𝐵 =< 𝐵′,
< 𝐶 =< 𝐶′,
< 𝐷 =< 𝐷′
Scale Factor (𝒌)
The scale factor is a measure of the amount of enlargement or reduction of the object and can be expressed as
an integer, a fraction or as a scale ratio.
2
𝑘 = 2 = = 2: 1
1
To determine the scale factor we use the equation
length of image
𝐴′𝐵′ 𝐵′𝐶′ 𝐶′𝐷′ 𝐷′𝐴′
𝑘=
=
=
=
=
length of original
𝐵𝐶
𝐴𝐵
𝐶𝐷
𝐷𝐴
*NOTE* If 𝑘 = 1 then the images will be the same size and this is called congruent
Exercise 7E – 1, 2, 3, 6
Similar Triangles
Triangles are considered similar if the satisfy the following conditions
 All three corresponding angles are equal
 All three corresponding pairs of sides are in the same ratio (all sides have been enlarged or reduced
by the same scale)
 Two sides of the triangles have been enlarged or reduced by the same scale and the angle between
these two side is the same on both triangles
To find the scale factor
𝑂𝐴′
length of side of image
𝑘=
=
𝑂𝐴 length of corresponding side of original
Similar triangles can have many practical applications. They are particularly useful to determine the length of
an unknown, difficult to measure or inaccessible distance or height.
Exercise 7F – 1a), c), d), f), 2b), e), 3, a), c), e), 4, 5, 6, 8
Area and Volume Scale Factors
If the lengths of similar figures are in ratio then it stands to reason that the area and volume of the similar
figures will be in ratio as well.
If lengths are of the ratio 𝑎: 𝑏 or 𝑘 then the areas will be in the ratio 𝑎2 : 𝑏 2 or 𝑘 2
area of image
Area scale ratio/factor (asf) =
= square of linear scale factor (lsf)=𝑘 2
area of orginal
If the lengths are of the ratio 𝑎: 𝑏 or 𝑘 then the areas will be in the ratio 𝑎3 : 𝑏 3 or 𝑘 3
Volume of image
Volume scale ratio/factor (vsf) =
= cube of linear scale factor (lsf)=𝑘 3
Volume of orginal
Exercise 7G – 1, 2a), d), 3, 4b), c), 5, 11, 14