Discover Mathematics P4: Term 2, Unit 13, Lessons 1–4
GoalFind the angle between the hands of a clock at an o’clock time, and estimate angles on the clock face.
New words
clock face · the round part of a clock with the numbers 1 to 12hour hand · the short hand that shows the hourminute hand · the long hand that shows the minutes
Explanation
A clock face is a circle with 12 equal spaces for the hours. A full turn is 360°, so each hour space is 360° ÷ 12 = 30°. At an o’clock time the minute hand points to 12. The angle between the hands is the number of hours from 12 to the hour hand, times 30°. At 3 o’clock: 3 × 30° = 90°, a right angle. At 6 o’clock: 6 × 30° = 180°, a straight angle. At 9 o’clock we count the other way, from the 9 up to the 12: 3 spaces, so 3 × 30° = 90° again. We always give the smaller angle. To estimate an angle on a clock when the hand is between two numbers, count the spaces and use the benchmark: half-way between two hour marks is 15° more than the first.
Worked examples
What is the angle between the hands at 4 o’clock? The hour hand is 4 spaces from 12, so the angle is 4 × 30° = 120°. This angle is bigger than 90° and smaller than 180°, so it is an obtuse angle.
What is the angle between the hands at 9 o’clock? Count the shorter way, from the 9 up to the 12. It is 3 spaces: 3 × 30° = 90°. Counting the long way would give 9 × 30° = 270°, which is not the angle we want.
Estimate at 2:30. The minute hand points to 6. The hour hand is half-way between the 2 and the 3. From the hour hand to the 6 is 3 spaces and a half: 3.5 × 30° = 105°. The angle is a little more than a right angle, so it is obtuse.
Class activity
Clock angles. Use a toy clock or draw a clock face with the numbers. Set the hands to 1, 2, 3, … 6 o’clock and write the angle between them each time. What do you notice about the angles? (They go up by 30° each hour.) Then find the angle at 7, 8 and 9 o’clock using the shorter way.
Practice
1
What is the angle between the hands of a clock at 2 o’clock?
2 × 30° = 60°.
2
What is the angle between the hands at 5 o’clock?
5 × 30° = 150°.
3
What is the smaller angle between the hands at 8 o’clock?
From the 8 up to the 12 is 4 spaces: 4 × 30° = 120°.
4
At what o’clock time is the angle between the hands 180°?
At 6 o’clock: 6 × 30° = 180°.
5
Estimate the angle between the hands at 3:30 (the hour hand is half-way between 3 and 4; the minute hand is on 6).
From the hour hand to the 6 is 2 spaces and a half: 2.5 × 30° = 75°.
