Download Volumes, arcs, sectors, angles and similar shapes

SCHOLAR Study Guide
National 5 Mathematics
Assessment Practice
Topic 2: Volumes, arcs, sectors,
angles and similar shapes
Authored by:
Margaret Ferguson
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
Copyright © 2016 SCHOLAR Forum.
Members of the SCHOLAR Forum may reproduce this publication in whole or in part for
educational purposes within their establishment providing that no profit accrues at any stage,
Any other use of the materials is governed by the general copyright statement that follows.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means, without written permission from the publisher.
Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the
information contained in this study guide.
Distributed by the SCHOLAR Forum.
SCHOLAR Study Guide Assessment Practice Topic 2: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
Acknowledgements
Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and
created these materials, and to the many colleagues who reviewed the content.
We would like to acknowledge the assistance of the education authorities, colleges, teachers
and students who contributed to the SCHOLAR programme and who evaluated these materials.
Grateful acknowledgement is made for permission to use the following material in the
SCHOLAR programme:
The Scottish Qualifications Authority for permission to use Past Papers assessments.
The Scottish Government for financial support.
The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA)
curriculum.
All brand names, product names, logos and related devices are used for identification purposes
only and are trademarks, registered trademarks or service marks of their respective holders.
1
Topic 1
Volumes, arcs, sectors, angles and
similar shapes
Contents
2.1
Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Assessment practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2
TOPIC 1. VOLUMES, ARCS, SECTORS, ANGLES AND SIMILAR SHAPES
By the end of this topic, you should have identified your strengths and areas for further
revision. Read through the learning points before you attempt the assessments and go
back to the Course Materials unit if you need more help.
You should be able to:
•
calculate the volume of a sphere, cone, pyramid, prism;
•
calculate an unknown dimension given the volume of a solid;
•
calculate the volume of composite solids;
•
calculate the length of an arc;
•
calculate the radius, diameter or angle given the length of the arc;
•
calculate the area of a sector;
•
calculate the radius, diameter or angle given the area of the sector;
•
calculate the interior angles of a polygon;
•
use the relationship in a circle between it’s centre, a chord and a perpendicular
bisector;
•
identify the relationship between a radius and a tangent to a circle;
•
identify the angle in a semi-circle;
•
identify similar shapes;
•
identify enlargement and reduction scale factors;
•
use a scale factor to find an unknown length;
•
work with similar triangles;
•
identify an area scale factor and use it to find an unknown area;
•
identify a volume scale factor and use it to calculate an unknown volume.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. VOLUMES, ARCS, SECTORS, ANGLES AND SIMILAR SHAPES
1.1
Learning points
Volumes of solids
•
Volume of a cuboid = lbh
•
Volume of a cylinder = πr 2 h
•
Volume of a cone = 13 πr 2 h
•
Volume of a pyramid =
•
Volume of a sphere = 43 πr 3
•
Volume of a prism = area of the constant cross-section × length
•
You may need:
◦
•
1
3
× Area of the base × height
Area of a Triangle = 12 bh
◦
Area of a Square = l 2
◦
Area of a Rectangle = lb
To find an unknown dimension on a solid shape when you know its volume:
◦
put the values you know into the formula including the volume
◦
rearrange or change the subject of the formula
Arcs and sectors
=
angle
360◦
× π × Diameter
•
Length of an Arc
•
Area of a Sector
•
To find an angle, radius or diameter when the length of an arc or the area of the
sector is known:
=
angle
360◦
× π × radius2
◦
put the values you know into the formula;
◦
re-arrange or change the subject of the formula.
Angles in a polygon
•
An angle at the centre of an n sided polygon = 360 ÷ n
•
An interior angle is an angle at a vertex of a polygon.
•
A star formed in a polygon will also have a smaller polygon within it.
Angle properties
•
Complementary angles add up to 90 ◦ .
•
Supplementary angles add up to 180 ◦ .
© H ERIOT-WATT U NIVERSITY
3
4
TOPIC 1. VOLUMES, ARCS, SECTORS, ANGLES AND SIMILAR SHAPES
•
Vertically opposite angles are equal (look for an "X" shape).
•
Corresponding angles are equal (look for an "F" shape).
•
Alternate angles are equal (look for a "Z" shape).
Angle properties in circles
•
A triangle formed by 2 radii and a chord is an isosceles triangle.
•
The angle in a semi-circle is a right angle.
•
A tangent is at right angles to the radius at the point of contact.
Similar shapes
•
Congruent shapes are an exact copy of each other.
•
Congruent shapes have a length scale factor of 1.
•
Similar shapes are an enlargement or reduction of each other.
•
If two shapes are similar then:
◦
pairs of corresponding angles are equal; and
◦
the ratios of pairs of corresponding sides are equal.
•
An enlargement scale factor is greater than 1 e.g. 5,
•
A reduction scale factor is less than 1 e.g. 1/2, 12 /25 .
•
If two triangles have equal angles they are equiangular.
22 / .
7
Similar triangles
•
If two triangles are similar then:
◦
pairs of corresponding angles are equal; or
◦
the ratios of pairs of corresponding sides are equal.
•
Triangles which are equiangular are similar.
•
In similar triangles:
◦
corresponding sides are opposite corresponding angles;
◦
there are corresponding, alternate and/or vertically opposite angles.
Area scale factor
•
Area scale factor = (length scale factor)2
Volume scale factor
•
Volume scale factor = (length scale factor)3
© H ERIOT-WATT U NIVERSITY
TOPIC 1. VOLUMES, ARCS, SECTORS, ANGLES AND SIMILAR SHAPES
1.2
5
Assessment practice
Make sure that you have read through the learning points or completed some revision
before attempting these questions. Tailor your practice by choosing the most appropriate
questions.
•
Volumes of solids: Questions 1, 2 and 3
•
Arcs and sectors: Questions 4, 5 and 6
•
Angles in polygons and circles: Questions 7, 8 and 9
•
Similar shapes: Questions 10, 11 and 12
Key point
Questions 2, 3, 4, 5, 6, 7, 8 and 9 also assess your reasoning skills.
Assessment practice: Volumes, arcs, sectors, angles and similar shapes
Volumes of solids
Go online
1
2
3 πr h
•
Volume of a cone =
•
Volume of a pyramid =
•
Volume of a sphere = 43 πr 3
•
Volume of a prism = area of the constant cross-section × length
1
3
× Area of the base × height
Q1: The Louvre Pyramid (Pyramide du Louvre) is a large glass and metal squarebased pyramid.
It’s height is approximately 20·6 m and the length of a side is 35 m.
Calculate the Volume of the pyramid.
Give your answer correct to 3 significant figures.
..........................................
© H ERIOT-WATT U NIVERSITY
6
TOPIC 1. VOLUMES, ARCS, SECTORS, ANGLES AND SIMILAR SHAPES
Q2: A shop is making glass ornaments with a Christmas tree inside.
The tree fits into a sphere of diameter 6·4 cm.
They decide to make similar decorations in the shape of a cone with height 7 cm and
the same volume.
Calculate the radius of the cone, correct to 3 significant figures.
..........................................
Q3: A cylinder of ice radius 0·6 m and height 1·4 m is placed in a triangular prism
shaped trough to melt.
The prism has width 1·5 m, height 1·2 m and length 1·7 m.
When the ice melts will the trough overflow?
Answer Yes or No and give a reason why.
..........................................
Length of an arc
A charity wants to make ribbons for an event.
A simplified ribbon is shown in Figure 1.
Q4: Calculate the length of the arc, correct to 3 significant figures.
The charity have a 50 m roll of pink ribbon.
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. VOLUMES, ARCS, SECTORS, ANGLES AND SIMILAR SHAPES
Q5: How many individual ribbons can they make from the roll?
..........................................
Area of a sector
Q6: A rear windscreen wiper moves across a sector of a circle.
Figure 1 shows that the angle the wiper arm makes is 118 ◦ and the wiper blade itself is
30 cm long.
Calculate the area of the windscreen which gets cleared.
Give your answer correct to 1 decimal place.
..........................................
Angles in polygons and circles
Q7: Calculate the size of an interior angle of this regular polygon.
..........................................
Q8: AB is a diameter of the circle with centre C. BT is a tangent at the point B.
Given that angle BTD = 30 ◦ , what is the size of angle CAD (in degrees)?
..........................................
© H ERIOT-WATT U NIVERSITY
7
8
TOPIC 1. VOLUMES, ARCS, SECTORS, ANGLES AND SIMILAR SHAPES
Q9:
The diagram shows a kite ABCD.
D is the centre of the circle and AB and BC are tangents to the circle at A and C
respectively.
Calculate the size of angle ABC.
..........................................
Similar shapes
Q10: Hammers come in various sizes and the two hammers below are similar.
Calculate the length of the large hammer.
..........................................
Q11: The chairs below are mathematically similar.
The large one is an adult size and the smaller one is a child size.
Calculate the area of fabric required for the child size if 0·85 m 2 of fabric is required for
the adult size.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. VOLUMES, ARCS, SECTORS, ANGLES AND SIMILAR SHAPES
..........................................
Q12: The two bottles of champagne are similar.
Calculate the cost of the large bottle.
..........................................
..........................................
© H ERIOT-WATT U NIVERSITY
9
10
ANSWERS: TOPIC 2
Answers to questions and activities
2 Volumes, arcs, sectors, angles and similar shapes
Assessment practice: Volumes, arcs, sectors, angles and similar shapes (page 5)
8410 m3
Q1:
Q2:
Steps:
•
Volume of sphere = 137·26
•
This is also the volume of the cone.
Answer: 4·33 cm
Q3:
Steps:
•
Volume of cylinder = 1·58
•
Volume of prism = 1·53
•
Yes. Volume of prism is 1·53 which is 0·05 cm 3 less than the volume of the cylinder
so when the cylinder melts the prism will overflow.
Answer: Yes
Q4:
4·40
Q5:
Steps:
•
50 ×100 = 5000 cm
•
5000 ÷ 13.9 = 359 ribbons
Answer: 359
Q6:
Hint:
•
Find the un-shaded area in the picture.
Steps:
•
Area of large sector = 2178·9
•
Area of small sector = 263·6
•
Use your answers to calculate the area cleared by the wiper blade (i.e. subtract).
Answer: 1915·3
Q7:
Steps:
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 2
11
•
Calculate the angle at the centre (360 ÷ n)
•
Calculate the interior angle.
Answer: 180 − (360 ÷ n)◦
For this heptagon, the interior angle = 128·6
Q8:
Steps:
•
Remember, the sum of the angles in a triangle = 180 ◦ .
•
BT is a tangent at the point B and CB is a radius of the circle. What is the size of
angle CBT (in degrees)? 90 ◦
•
What is the size of angle BCT (in degrees)? 60 ◦
•
AB is a diameter what size is angle DCA? 120 ◦
Answer: ∠CAD = 30 ◦
Q9: 2(90 - 71) = 38◦
Q10:
Steps:
10·5
7·5
•
What is the length scale factor (do not round)?
•
Use this answer to find the length of the large hammer.
Answer: 30·8 cm
Q11:
Steps:
•
What is the length scale factor (do not round)?
•
What is the area scale factor (do not round)?
•
Use this answer to find the area of fabric required for the child’s chair.
68 /
86
68
( /86 )2
Answer: 0·53 m2
Q12:
Steps:
•
What is the length scale factor (do not round)?
•
What is the volume scale factor (do not
•
Use this answer to find the area of fabric required for the child’s chair.
Answer: £33·05
© H ERIOT-WATT U NIVERSITY
31 /
26
31
round)? ( /26 )3