RIC-6835 Maths Essentials - Geometry and Measurement 1 (Ages 11-15)

Equivalent measures
Capacity
Length
1 000 000 cm 3 = 1 m 3
10 mm = 1 cm
100 cm = 1 m
1000 m = 1 km
Time
Area
100 mm 2 = 1 cm 2
10 000 cm 2 = 1 m 2
10 000 m 2 = 1 ha
100 ha = 1 km 2
Mass
1000 mg = 1 g
1000 g = 1 kg
1000 kg = 1 t
Volume
1000 mm 3 = 1 cm 3
Angles of elevation
The angle of elevation is the angle ray AB
makes with the horizontal.
Angles of depression
The angle of depression is the angle ray CD
makes with the horizontal.
Trigonometric ratios
How to choose a trigonometric ratio:
For right angled triangles:
θ
Hypotenuse
Adjacent
Opposite
1. Label each side of the triangle with O, A and H.
2. Decide which two sides are part of the problem.
3. Use: SOH sin θ = O
H or
CAH cos θ = A
H or
O
TOA tan θ =
A
to solve the problem.
Measurement
Measurement
1000 mL = 1 L
1000 L = 1 kL
60 seconds = 1 minute
60 minutes = 1 hour
24 hours = 1 day
7 days = 1 week
14 days = 1 fortnight
28–31 days = 1 month
~ 52 weeks = 1 year
12 months = 1 year
365 days = 1 year
366 days = 1 leap year
10 years = 1 decade
100 years = 1 century
1000 years = 1 millennium
Unknown side
Label it with a pronumeral (x) and
calculate.
15º
x mm (H)
For example:
20
sin 15° = x
20
x = sin 15°
x = 77.3 mm
20 mm
(O)
Decimal measures
Decimal measures
Measurements should be written in decimal
form where possible.
For example:
10 cm, 3 mm 10.3 cm
2 m, 7 cm, 5 mm 2.075 m
3 km, 408 m, 52 cm 3.40852 km
6 L, 430 mL 6.430 L
15 kL, 675 L 15.675 kL
12 g, 325 mg 12.325 g
4 kg, 75 g 4.075 kg
5 t, 620 kg 5.620 t
Temperature
A thermometer is used to measure
temperature.
Freezing
point of
water:
0 °C
Measuring tools
Various tools are used to measure in
different situations. For example:
length
mass
capacity
Unknown angle
Choose the trigonometric ratio
using the steps shown and calculate.
For example:
θº
215 cm (A)
Boiling
point of
water:
100 °C
ruler
tape measure
trundle wheel
balance
kitchen scales
bathroom scales
measuring spoons
measuring cups
measuring jugs
graduated cylinders
tan θ = O A
84
= 215
θ = tan ( 215)
–1 84
θ = 21.3°
84 cm
(O)
2-D shapes
2-D shapes have two dimensions—width and length.
They may have curved or straight sides.
Shapes with straight sides are called
‘polygons’. Polygons have three or more
sides and angles.
Polygons with sides of an equal length and
angles of an equal size are called regular
polygons.
Polygons with three sides are called ‘triangles’.
3-D shapes
Parts of a 3-D shape include:
faces
The surfaces of a three-dimensional fi gure.
edges
The intersections of two faces of a
three-dimensional fi gure.
vertices
The intersections of three edges of a threedimensional
fi gure.
Geometry
Geometry
Polygons with four sides are called
‘quadrilaterals’.
Quadrilaterals with opposite sides parallel
are called ‘parallelograms’.
Other shapes include circles, semicircles and
ellipses.
prisms
If the two ends of a prism are
the same size and shape; they
are congruent.
Nets
A net is a 2-D plan which can be used to
make a 3-D shape.
Three-dimensional shapes are also called ‘solid figures’. A 3-D shape has length, width
and height.
©R.I.C. Publications
Low Resolution Images
Display Copy
regular polyhedra
The faces of these 3-D shapes are all the same shape
and size. There are fi ve: tetrahedron, octahedron,
hexahedron, icosahedron and dodecahedron.
pyramids
A pyramid is made up of a base shape such as a
triangle, square, hexagon or decagon.
The opposite end of the shape forms a point
called an ‘apex’.
Angles
An angle (
acute angle
An angle less than 90°.
right angle
An angle of exactly 90°.
obtuse angle
An angle between
90° and 180°.
reflex angle
An angle between
180° and 360°.
The four angles in a
quadrilateral add up to
360º.
rectangle
90º + 90º + 90º + 90º = 360º
rhombus
a + b + a + b = 360°
parallelogram
a + b + a + b = 360º
acute
angle
right
angle
obtuse angle
reflex angle
straight angle
An angle which is exactly
180°.
straight angle
complementary
angles
Two angles which add to
90° (a right angle).
) is formed when two lines meet. Angles are measured in degrees.
A degree ( ) is a unit of angular measure. There are 360° in one complete
rotation.
A protractor is a tool used to measure the size of an angle.
Quadrilaterals
50°
40°
supplementary angles
Two angles which
add to 180° (a
straight angle).
corresponding
angles
alternate angles
co-interior
angles
Geometry
Geometry
50°
130°
vertically opposite angles
Congruent angles formed when two lines
intersect.
Mathematical terms
attribute
A characteristic of an object. A
way to classify objects; e.g. round,
red, thick.
classification
Arrangement into classes
(sets or groups) according to
attributes.
congruent
Two fi gures are congruent if they
are the same size and shape.
coordinates
A referenced point on a grid can
be found using an ‘ordered pair’
of numbers. These are called the
coordinates of the point. The
horizontal axis is always read or
written before the vertical axis.
line
Made up of an infi nite set
of points extending in both
directions.
line segment
A line with two end points.
model
A representation of an object
preserving the signifi cant
features.
net
A fl at pattern that can be folded
to make a three-dimensional
model.
network
A system of lines (paths) and
nodes (points representing
intersections).
parallel lines
Two or more straight lines in
the same plane which will never
meet. They are always the same
distance apart.
path
A line connecting points (nodes)
in a network.
perpendicular lines
Two lines which form right angles.
plane
A surface which is fl at and has no
boundaries.
Triangles
Triangles
angles in a triangle
Triangles can be named according to the size
of their angles.
exterior angles
The exterior angle
of a triangle is
equal to the sum of
the two opposite
interior angles.
congruent triangles
Triangles are congruent if they are identical
in shape and size.
Two triangles can be called congruent if:
SSS: all sides the same length or
SAS: two sides the same and the angle
between those sides the same or
ASA: two angles the same and the side
between those angles the same length
or
RHS: If the hypotenuse and a side of one
triangle are respectively equal to the
hypotenuse and a side of the other
triangle, then the two right triangles
are congruent.
ray
Made up of an infi nite set of
points emanating from a point
and going in one direction.
scale
The ratio of measurements
of a model or diagram to
corresponding measurements of
an enlarged or reduced version.
section
A fl at surface made by cutting
through a solid in any direction.
similar
Two shapes are similar when
they have the same shape but are
different in size.
tessellation
A repeating pattern of congruent
shapes that completely cover an
area leaving no gaps or overlaps.
traversable
A network is traversable if all
paths can be traced over without
going over the same path twice.
Maths
No part of this publication may be reproduced in any form or
by any means, electronic or mechanical, including photocopying
or recording, or by any information storage and retrieval
system, without written permission from the publisher.
RIC–6835
Copyright Information
9!BMFBI