Constructing a 45° angle

The idea

A 45° angle is constructed with ruler and compass by first constructing a 90° angle and then bisecting it.

Eight lines for a design

A design has 8 lines through one point O. Every angle between neighbouring lines is equal, and together they fill the whole turn of 360°.

How many degrees is each angle?

• 45°
• 60°
• 90°

Make the eight lines

Start with one line. First make a 90° angle at O. Then bisect each 90° angle by tapping it.

A right angle cut in half

You already know how to construct a 90° angle (Lesson 5) and how to bisect an angle (Lesson 6). Bisecting a right angle gives two equal angles: 90° ÷ 2 = 45°. Four right angles make 8 angles of 45°.

A 45° angle is constructed by first constructing a 90° angle and then bisecting it.

• Step | Angle you get
• Construct a right angle at O | 90°
• Bisect the 90° angle | 45° and 45°
• Bisect all four right angles | eight angles of 45°

Notes

A 45° angle is constructed by first constructing a 90° angle and then bisecting it. Eight such angles fill the whole turn.

Check yourself

Asha makes a 90° angle and bisects it. She joins one of the 45° angles to the 90° angle next to it. How many degrees is the new big angle?

Answer: 135

The big angle is 90° + 45° = 135°.

Meena wants 45° but bisects a 60° angle. What does she get?

How many 45° angles fit side by side in a straight line (180°)?

Answer: 4

180° ÷ 45° = 4.

• Two angles of 45°.. Half of 60° is 30°, not 45°. Half of 90° is 45°.
• Two angles of 30°. A 60° angle must be bisected to get 30°. — correct. Yes! 60° ÷ 2 = 30°, so 45° needs the 90° angle.
• Two angles of 60°.. A bisector cuts the angle into halves, so each part is smaller than 60°.