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Angles

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Related Angles

Lines AB and CD are parallel to one another (hence the »

on the lines).

a and d are vertically opposite angles. Vertically opposite angles

are equal. (b and c, e and h, f and g are also vertically opposite).

g and c are corresponding angles. Corresponding angles are equal. (h

and d, f and b, e and a are also corresponding).

d and e are alternate angles. Alternate angles are equal. (c and f

are also alternate). Alternate angles form a 'Z' shape and are sometimes

called 'Z angles'.

a and b are adjacent angles. Adjacent angles add up to 180 degrees.

(d and c, c and a, d and b, f and e, e and g, h and g, h and f are also

adjacent).

d and f are interior angles. These add up to 180 degrees (e and c

are also interior).

Any two angles that add up to 180 degrees are known as supplementary

angles.

The angles around a point add up to 360 degrees.

The angles in a triangle add up to 180 degrees.

The angles in a quadrilateral add up to 360 degrees.

The angles in a polygon (a shape with n sides) add up to 180(n - 2)

degrees.

The exterior angles of any polygon add up to 360 degrees.

Areas and Volumes

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*** Remember, with many exam boards, formulae will be

given to you in the exam. However, you need to know how to apply the

formulae and learning them (especially the simpler ones) will help you in

the exam. ***

A prism is a shape with a constant cross section, in other words the

cross-section looks the same anywhere along the length of the solid

(examples: cylinder, cuboid). The volume of a prism = the area of the

cross-section × the length. So, for example, the volume of a cylinder =

pr² × length.

Areas :

The area of a triangle = half × base × height

The area of a circle = pr²

(r is the radius of the circle)

The area of a parallelogram = base × height

Area of a trapezium = half × (sum of the parallel sides) × the distance

between them [ 1/2(a+b)d ].

Spheres:

Volume: 4/3pr³

Surface area: 4pr²

Cylinder:

Curved surface area: 2prh

Volume: pr²h

Pyramid:

Volume = 1/3 × area of base × perpendicular height (=1/3pr²h for circular based pyramid).

Cone:

Curved surface area: prl (l is

slant height)

Volume: 1/3pr²h (h is perpendicular

height)

WHEN USING FORMULAE FOR AREA AND VOLUME IT IS NECESSARY THAT ALL

MEASUREMENTS ARE IN THE SAME UNITS.

Units

1 kilometre (km) = 1000 m

1 metre (m) = 100cm

1 centimetre (cm) = 10mm

1 litre = 1000 cm³

1 hectare = 10 000 m²

1 kilogram (kg) = 1000g (grams)

When working with lengths try to use metres if possible and when working

with mass, use kilograms.

1cm² = 100mm² (10mm × 10mm)

1cm³ = 1000mm³ (10mm × 10mm × 10mm)

Ratios of lengths, areas and volumes

Imagine two squares, one with sides of length 3cm and one with sides of

length 6cm. The ratio of these lengths is 3 : 6 (= 1 : 2). The area of the

first is 9cm and the area of the second is 36cm. The ratio of these areas

is 9 : 36 (= 1 : 4) .

In general, if the ratio of two lengths (of similar shapes) is a : b, the

ratio of their areas is a² : b² . The ratio of their volumes is a³ :

b³ .

This is why the ratio of the length of a mm to a cm is 1:10 (there are 10mm

in a cm). The ratio of their areas (i.e. mm² to cm²) is 1:10² (there are

100mm² in a cm²) and the ratio of their volumes (mm³ to cm³) is 1:10³

(there are 1000mm² in a cm²).

Dimensions

Lines have one dimension, areas have two dimensions and volumes have three.

Therefore if you are asked to choose a formula for the volume of an object

from a list, you will know that it is the one with three dimensions.

Example:

The letters r, l, a and b represent lengths. From the following, tick the

three which represent volumes.

pr²l

2pr²

4pr³

abrl

abl/r

3(a² + b²)r

prl

NB: Numbers are dimensionless so ignore p,

2, 4 and 3.

The first has three dimensions, since it is r × r × l.

The second has two dimensions (r × r).

The third has three dimensions (r × r × r).

etc.

3(a² + b²)r is the third formula with three dimensions. The expanded

version of this formula is 3a²r + 3b²r and 3 dimensions + 3 dimensions = 3

dimensions (the dimension can only be increased or reduced by

multiplication or division).

© Matthew Pinkey

Circle Theorems

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Circles

The red line in the second diagram is called a chord. It

divides the circle into a major segment and a minor segment.

The diagrams show that:

a) The angle formed at the centre of the circle by lines originating from

two points on the circle's circumference is double the angle formed on the

circumference of the circle by lines originating from the same points. i.e.

in the first diagram, a = 2b.

b) Angles formed from two points on the circumference are equal to other

angles, in the same arc, formed from those two points.

c) Angles formed by drawing lines from the ends of the diameter of a circle

to its circumference form a right angle. So in the third diagram, c is a

right angle.

d) A tangent to a circle forms a right angle with the circle's radius, at

the point of contact of the tangent (a tangent to a circle is a line that

touches the circumference at one point only).

e) The final diagram shows the alternate segment theorem. In short, the red

angles are equal to each other and the green angles are equal to each

other.

A cyclic quadrilateral is a four-sided figure in a

circle, with each vertex (corner) of the quadrilateral touching the

circumference of the circle. The opposite angles of such a quadrilateral

add up to 180 degrees.

Area of Sector and Arc Length

If the radius of the circle is r,

Area of sector = pr² × A/360

Arc length = 2pr × A/360

In other words, area of sector = area of circle × A/360

arc length = circumference of circle × A/360

Loci

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Loci

A locus is a set of points satisfying a certain condition. The term

'locus', however, is rarely used in exams. The question is more likely to

be of this format:

Example:

The diagram shows two points P and Q. On the diagram shade the region which

contains all the points which satisfy both the following: the distance from

P is less than 3cm, the distance from P is greater than the distance from

Q.

All of the points on the circumference of the circle are

3cm from P. Therefore all of the points satisfying the condition that the

distance from P is less than 3cm are in the circle.

If we draw a line in the middle of P and Q, all of the points on this line

will be the same distance from P as they are from Q. They will be therefore

closer to Q, and further away from P, if they are on the right of such a

line.

Therefore all of the points satisfying both of these conditions are shaded

in red.

Shapes

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Symmetry

If a shape has a line of symmetry, the line of symmetry will divide the

shape into two equal parts, one half of which can be folded along the line

of symmetry to fit exactly onto the other. Note, a rectangle has two (not

four) lines of symmetry and a circle has an infinite number.

If rotating a shape through a certain angle produces an identical shape, it

has rotational symmetry. If the shape can be rotated 4 times before

returning to its original shape (e.g. a square), it has rotational symmetry

of order 4. An equilateral triangle has rotational symmetry of order 3 and

a rectangle of order 2.

Triangles

Isosceles triangles have two equal angles. The sides of the triangle

opposite the equal angles are equal in length to one another.

Equilateral triangles have all of their sides and angles equal. Since there

are 180 degrees in a triangle and all the angles are equal, each angle is

60 degrees.

Other Shapes:

Parallelogram: opposite sides are parallel, opposite

angles are equal, the diagonals bisect one another.

Rhombus: (a parallelogram with all four sides of equal length), diagonals

bisect one another at right angles.

Trapezium: One pair of opposite sides are parallel.

Square: All sides are equal, all angles are 90 degrees, diagonals bisect

one another at 90 degrees.

Rectangle: All angles are 90 degrees, diagonals bisect one another.

angles

areas and volumes

circle theorems

loci

shapes