Measuring and Drawing Lines and Angles: Edexcel GCSE Foundation maths revision notes

Measuring and drawing in the exam

This page is about measuring lines with a ruler and angles with a protractor, and drawing them accurately. Take a ruler marked in centimetres and millimetres, and a protractor, to every paper.

In the exam you are usually asked to "Measure the length of" a line or a side of a triangle, and to "Measure the size of the angle marked x", for 1 mark each. You may also be asked what type of angle it is.

Only measure when the question says the diagram is accurately drawn. Most diagrams in the exam are not, and then you work angles and lengths out instead.

A screen changes the size of a picture, so you can't use a real ruler on the diagrams here. The examples show you how to read the scales instead.

Measuring a line with a ruler

A ruler is marked in centimetres (cm). Each centimetre is split into 10 small marks, and each small mark is a millimetre (mm). The longer mark halfway between two centimetres is 5 mm.

1 \text{ cm} = 10 \text{ mm}

Put the 0 on the ruler at one end of the line. The 0 is not always at the very end of the ruler. Then read the other end of the line: count the whole centimetres, then the small marks after them.

Measure to the nearest millimetre: 4.3 cm, not 4 cm. To change centimetres into millimetres, multiply by 10, so 4.3 cm = 43 mm. To change millimetres into centimetres, divide by 10

Worked example 1: Reading a ruler

The diagram shows a line drawn above a ruler. Write down the length of the line in centimetres, and then in millimetres.

The line starts at 0 on the ruler, so its length is the reading at the other end.
The other end is past the 7 cm mark but not as far as 8, so the line is 7 cm and some millimetres.
Your turn. Count the small marks after 7. Write down the length of the line in centimetres.

Your answer

The end is 4 small marks past 7, so the line is 7.4 cm
In millimetres: 7.4 \times 10 = 74 mm
Answer: 7.4 cm, which is 74 mm

Worked example 2: A line that does not start at 0

A line is drawn above a ruler. Its left end is not at 0. Work out the length of the line. Give your answer in millimetres.

The line starts at 1.7 cm and ends at 7.3 cm on the ruler.
The length is the gap between the two readings, so subtract the start from the end. Reading only the end, 7.3 cm, would be wrong.
Your turn. Work out 7.3 - 1.7

Your answer

7.3 - 1.7 = 5.6 cm
Check: count on from the start, 1.7 + 5.6 = 7.3
Change to millimetres: 5.6 \times 10 = 56 mm
Answer: 56 mm

Types of angle

An angle is an amount of turn, measured in degrees (^\circ). A full turn is 360^\circ, a straight line is 180^\circ and a right angle is 90^\circ

Acute: less than 90^\circ
Right angle: exactly 90^\circ
Obtuse: more than 90^\circ but less than 180^\circ
Reflex: more than 180^\circ but less than 360^\circ

To decide the type of an angle, compare it with a right angle and a straight line. The length of the two lines makes no difference: a small angle with long lines is still small.

Worked example 3: Naming the type of angle

Here are five angles: 35^\circ, 90^\circ, 128^\circ, 179^\circ and 205^\circ. Write down the type of each angle.

Compare each angle with 90^\circ, a right angle, and 180^\circ, a straight line.
35^\circ is less than 90^\circ, so it is acute. 90^\circ is exactly a right angle, so it is neither acute nor obtuse.
Your turn. Obtuse angles are more than 90^\circ but less than 180^\circ. Which angles in the list are obtuse? Separate them with commas.

Your answer

Give every value, separated by commas

128^\circ and 179^\circ are each more than 90^\circ and less than 180^\circ, so they are obtuse
205^\circ is more than 180^\circ, so it is reflex. An obtuse angle is never more than 180^\circ
Answer: 35^\circ is acute, 90^\circ is a right angle, 128^\circ and 179^\circ are obtuse, and 205^\circ is reflex.

Measuring an angle with a protractor

A protractor has two scales, each going from 0 to 180. One starts at 0 on the right and the other starts at 0 on the left, so every mark has two numbers. A real protractor has a mark for every degree. The protractors drawn on this page have a mark every 10^\circ, so they are easier to read.

Put the centre of the protractor exactly on the corner of the angle.
Line up the 0 line with one of the two lines of the angle.
Use the scale that starts at 0 on that line.
Follow it round and read where the other line crosses it.

The angle is acute, so it is 60^\circ, not 120^\circ. Decide whether an angle is acute or obtuse before you read it: that tells you which number is right. The two numbers at any mark add up to 180

Angles are often named with three letters. Angle ABC is the angle at B, between the lines BA and BC: the middle letter is the corner. Put the centre of the protractor on that letter.

Worked example 4: Choosing the right scale

The diagram shows a protractor placed on angle ABC. Write down the size of angle ABC.

The centre of the protractor is on B, the middle letter. Line BA lies along the 0 line on the left.
Angle ABC is wider than a right angle, so it is obtuse and the answer is more than 90^\circ
The scale that starts at 0 on the left, at line BA, is the outer scale. Follow it round: 30, 60, 90, 120, …
Your turn. Each small mark is 10^\circ. Read the outer scale where line BC crosses it. What is the size of angle ABC?

Your answer

Line BC crosses the outer scale one small mark past 120, so angle ABC = 130^\circ
Check: the inner scale reads 50 there, but 50^\circ is acute, so it is the wrong scale. And 130 + 50 = 180
Answer: 130^\circ

Measuring a reflex angle

A protractor only goes up to 180^\circ, so you can't read a reflex angle straight off it.

Measure the smaller angle between the two lines instead. The two angles make a full turn, so subtract the smaller angle from 360^\circ

\text{reflex angle} = 360^\circ - \text{smaller angle}

Here r = 360 - 70 = 290^\circ. Check: 290^\circ is more than 180^\circ and less than 360^\circ, so it is reflex.

Worked example 5: A reflex angle

The diagram shows a protractor placed on an angle. Angle y is the reflex angle, marked the long way round. Work out the size of angle y.

The protractor measures the smaller angle between the two lines. One line is on the 0 of the inner scale, on the right.
The other line crosses the inner scale one small mark past 30, so the smaller angle is 40^\circ
Angle y and the 40^\circ angle make a full turn, 360^\circ
Your turn. Work out 360 - 40

Your answer

y = 360 - 40 = 320^\circ
Answer: y = 320^\circ

Drawing lines and angles

To draw a line 6.3 cm long, put the 0 of the ruler where the line starts. Make a small mark 3 small marks past 6, then join the two points with the ruler. Use a sharp pencil.

Draw one of the two lines with a ruler, and label the corner.
Put the centre of the protractor on the corner, with the 0 line along that line.
Use the scale that starts at 0 on that line. Count round to the angle you want and make a dot.
Take the protractor away and draw a line from the corner through the dot.

Check by eye: 70^\circ is acute, so your angle should be smaller than a right angle. If it looks obtuse, you have used the other scale and drawn 110^\circ

Worked example 6: Marking the midpoint of a line

The diagram shows a line AB drawn above a ruler. Work out where on the ruler the midpoint of AB is, so that you can mark it with a cross.

The midpoint of AB is the point halfway along it. A is at 1.2 cm on the ruler and B is at 7.8 cm.
Length of AB: 7.8 - 1.2 = 6.6 cm
The midpoint is half the length of AB from A: 6.6 \div 2 = 3.3 cm
Your turn. Count on 3.3 cm from A. At what reading on the ruler is the midpoint?

Your answer

1.2 + 3.3 = 4.5, so the midpoint is at 4.5 cm on the ruler
Check: from the midpoint to B is 7.8 - 4.5 = 3.3 cm, the same as from A to the midpoint.
Answer: mark the cross at 4.5 cm on the ruler, 3.3 cm from each end of AB.

Common mistakes

Reading the wrong scale on the protractor. Decide first whether the angle is acute or obtuse, then use the scale that starts at 0 on one of the two lines.
Starting from the end of the ruler, or from 1. Put the 0 at the start of the line, or read both ends and subtract.
Rounding a length to a whole centimetre. Count the small marks too, and give an answer such as 8.6 cm.
Mixing up centimetres and millimetres. Give the length in the unit the question asks for: 86 mm is 8.6 cm.
Mixing up the types of angle. Acute is less than 90^\circ, obtuse is between 90^\circ and 180^\circ, and reflex is more than 180^\circ
Reading a reflex angle straight off the protractor. Measure the smaller angle, then subtract it from 360^\circ