How To Draw A 90 Degree Angle Without A Protractor

In this section, we will consider the construction of some angles with special sizes.

Constructing a 60� Angle

We know that the angles in an equilateral triangle are all 60� in size. This suggests that to construct a 60� angle we need to construct an equilateral triangle as described below.

Step 1: Draw the arm PQ.
Step 2: Place the point of the compass at P and draw an arc that passes through Q.
Step 3: Place the point of the compass at Q and draw an arc that passes through P. Let this arc cut the arc drawn in Step 2 at R.

Constructing a 30� Angle

We know that:

So, to construct an angle of 30�, first construct a 60� angle and then bisect it. Often, we apply the following steps.

Step 1: Draw the arm PQ.
Step 2: Place the point of the compass at P and draw an arc that passes through Q.
Step 3: Place the point of the compass at Q and draw an arc that cuts the arc drawn in Step 2 at R.
Step 4: With the point of the compass still at Q, draw an arc near T as shown.
Step 5: With the point of the compass at R, draw an arc to cut the arc drawn in Step 4 at T.
Step 6: Join T to P. The angle QPT is 30�.

Constructing a 120� Angle

We know that:

This means that 120� is the supplement of 60�. Therefore, to construct a 120� angle, construct a 60� angle and then extend one of its arms as shown below.

Constructing a 90� Angle

We can construct a 90� angle either by bisecting a straight angle or using the following steps.

Step 1: Draw the arm PA.
Step 2: Place the point of the compass at P and draw an arc that cuts the arm at Q.
Step 3: Place the point of the compass at Q and draw an arc of radius PQ that cuts the arc drawn in Step 2 at R.
Step 4: With the point of the compass at R, draw an arc of radius PQ to cut the arc drawn in Step 2 at S.
Step 5: With the point of the compass still at R, draw another arc of radius PQ near T as shown.
Step 6: With the point of the compass at S, draw an arc of radius PQ to cut the arc drawn in step 5 at T.
Step 7: Join T to P. The angle APT is 90�.