Constructing Angles of 90° and 45°

1. The right angle is everywhere

Look at the corner of your classroom, the corner of this page, the corner of a window frame or a Ghana Cedi note. All of them make a right angle, which measures exactly 90°. Builders need right angles so that walls stand straight. If a corner is even 2° out, a long wall drifts a long way off line.

A protractor or a set square can give you a 90° angle quickly, but they can be inaccurate or unavailable. In this lesson you will build a 90° angle using only a pair of compasses and a ruler. Then you will cut it in half to get 45°. Half of 90° is 45°, so a 45° angle is simply a bisected right angle.

2. Constructing an angle of 90°

The syllabus asks you to raise a perpendicular at a point on a line, which is the same as building a 90° angle. We build it from the 60° and 120° angles that you already know how to make with equal arcs. The key idea is that 90° is exactly halfway between 60° and 120°.

1. 1Draw a straight line. Mark the point O on it where the angle will be.
2. 2With the compass point on O, draw a large arc that cuts the line at A.
3. 3Keep the same opening. Put the compass point on A and cut the arc at B. Angle AOB = 60°.
4. 4Keep the same opening. Put the compass point on B and cut the arc again at C. Angle AOC = 120°, so angle BOC = 60°.
5. 5From B and then from C (same opening, any convenient width), draw arcs that cross at D. D is exactly between B and C.
6. 6Join O to D. Angle AOD = 60° + 30° = 90°.

▶ Notice that BOC is a 60° angle. Bisecting it gives 30°, and 60° + 30° = 90°.

Check with a protractor: place the centre of the protractor on O and the baseline along OA. The line OD should pass through 90° (within 1°).

Another way: a perpendicular at a point. A line PQ is perpendicular to AB at P if the angle at P is 90°. Put the compass point on P and cut the line at A and B so that PA = PB. Open the compass wider and draw arcs from A and from B above P so they cross at Q. Join P and Q. This is just the perpendicular bisector method applied to the segment AB, because P is the midpoint of AB.

▶ Because PA = PB, P is the midpoint of AB. The line PQ is its perpendicular bisector, so angle QPB = 90°.

3. Constructing an angle of 45°

Forty-five degrees is half of a right angle, so we begin with the 90° construction and then bisect it. The bisecting steps are the ones you used in the last lesson: an arc that cuts both arms and two equal arcs that cross.

1. 1Construct a 90° angle AOD as shown above.
2. 2With the compass on O, draw an arc cutting OA at X and OD at Y.
3. 3With the compass on X and then on Y (same opening), draw arcs that cross at E.
4. 4Join O to E. Angle AOE = 90° ÷ 2 = 45°.

▶ First the 90° angle is built, then it is bisected. Notice that angle AOE and angle EOD are both 45°.

Fixing the arm lengths. Some exercises say: Construct angle AOB = 45° such that OA = 5 cm and OB = 6 cm. This simply fixes the length of the arms. After you construct the 45° angle, measure 5 cm along one arm from O to mark A, and 6 cm along the other arm to mark B. The angle does not change when the arms are longer or shorter.

4. Using right angles and 45° angles

Right angles and 45° angles are used in many trades. A tailor cuts cloth on the bias (45°) so that it stretches along the diagonal. A mason makes sure the foundation corner is 90°. A carpenter cuts a mitre joint at 45° to join two pieces of picture frame so that they form a 90° corner.

There is a very useful pattern. If you bisect a right angle you get 45°. If you bisect again you get 22.5°. Four right angles fit around a point (4 × 90° = 360°), and four 45° angles fit in a straight line (4 × 45° = 180°).

Worked examples

Try each one yourself first, then open the solution and compare every step.