Constructing 60∘60∘ angle using Compass

overview

In this page, constructing the following using a compass are explained.

• 60∘60∘ angle

• 30∘30∘ angle

• 120∘120∘ angle

• 15∘15∘ angle

• 90∘90∘ angle

Constructing 60∘ angle using Compass

We know that, Angles in a equilateral triangle are 60∘. This geometrical property helps to construct 60∘ angle using a compass.

To construct an angle measuring 60∘ at point A on line ¯AB, construct a equilateral triangle as given below.

• With an arbitrary distance measure on compass, mark point P from point A on line ¯AB.

• With the same distance measure on compass, construct two arcs from points A and P. The arcs cut at point Q.

Since the distance measure on compass is identical, points A, P, and Q make an equilateral triangle. The line ¯AQ is extended and ∠PAQ is 60∘

**Constructing 60∘ angle using a Compass** : Construct an equilateral triangle and the angle 60∘ is constructed at the vertex.

Constructing 30∘ angle using Compass

To construct 30∘ angle using a compass, we use the property that 30∘ is half of 60∘.

We have learned constructing 60∘ and bisecting an angle using a compass. Combine these two to construct 30∘ angle.

• With an arbitrary distance-measure on compass, mark point P from point A on line ¯AB.

• With the same distance measure on compass, construct two arcs from points A and P. The arcs cut at point Q. ¯AQ makes 60∘ with the ¯AP.

• With an arbitrary distance-measure on compass, construct two arcs from points P and Q. These arcs cut at point R. The line ¯AR bisects the angle ∠PAQ.

The angle 30∘ is constructed as ∠PAR.

**Constructing 30∘ angle using a Compass** : Construct an equilateral triangle and the angle 60∘ is constructed at the vertex. Bisect the angle.

Constructing 120∘ angle using Compass

To construct 120∘ angle using a compass, we use the property that 120∘ is double of 60∘.

Constructing 60∘ angle twice-over to construct 120∘

• With an arbitrary distance-measure on compass, mark point P from point A on line ¯AB.

• With the same distance measure on compass, construct two arcs from points A and P. The arcs cut at point Q. ¯AQ makes 60∘ with the ¯AP.

• With the same distance measure on compass, construct two arcs from points A and Q. The arcs cut at point R. ¯AR makes 120∘ with the ¯AP.
The angle 120∘ is constructed as ∠PAR.

**Constructing 120∘ angle using a Compass** : Construct an equilateral triangle and another equilateral triangle on the side of the constructed one.

Constructing 15∘ angle using Compass

To construct 15∘ angle using a compass, we use the property that 15∘ is half of 30∘.

We have learned constructing 60∘ and bisecting an angle using a compass. Combine these two to construct 15∘ angle.

• With an arbitrary distance-measure on compass, mark point P from point A on line ¯AB.

• With the same distance measure on compass, construct two arcs from points A and P. The arcs cut at point Q. ¯AQ makes 60∘ with the ¯AP.

• With an arbitrary distance-measure on compass, construct two arcs from points P and Q. These arcs cut at point R. The line ¯AR bisects the angle ∠PAQ.

• With an arbitrary distance-measure on compass, construct two arcs from points P and R. These arcs cut at point S. The line ¯AS bisects the angle ∠PAS.

The angle 15∘ is constructed as ∠PAS.

**Constructing 15∘ angle using a Compass** : Construct an equilateral triangle and the angle 60∘ is constructed at the vertex. Bisect the angle twice.

Constructing 90∘ angle using Compass

To construct 90∘ angle using a compass, we use the property that 90∘ is sum of 60∘ and 30∘.

Constructing 90∘ with two 60∘ is illustrated

• With an arbitrary distance-measure on compass, mark point P from point A on line ¯AB.

• With the same distance measure on compass, construct two arcs from points A and P. The arcs cut at point Q. ¯AQ makes 60∘ with the ¯AP.

• With the same distance measure on compass, construct two arcs from points A and Q. The arcs cut at point R. ¯AR makes 120∘ with the ¯AP.

• With an arbitrary distance-measure on compass, construct two arcs from points Q and R. These arcs cut at point S. The line ¯AS bisects the angle ∠QAR.

The angle 90∘ is constructed as ∠PAS.

**Constructing 90∘ angle using a compass** : Construct two equilateral triangles and bisect the second one to construct 90∘ angle.

summary

**Constructing 60∘ angle using a Compass** : Construct an equilateral triangle and the angle 60∘ is constructed at the vertex.

**Constructing 30∘ angle using a Compass** : Construct an equilateral triangle and the angle 60∘ is constructed at the vertex. Bisect the angle.

**Constructing 120∘ angle using a Compass** : Construct an equilateral triangle and another equilateral triangle on the side of the constructed one.

**Constructing 15∘ angle using a Compass** : Construct an equilateral triangle and the angle 60∘ is constructed at the vertex. Bisect the angle twice.

**Constructing 90∘ angle using a compass** : Construct two equilateral triangles and bisect the second one to construct 90∘ angle.

Outline

The outline of material to learn "Construction / Practical Geometry at 6-8th Grade level" is as follows.
Note: * click here for detailed outline of "constructions / practical geometry".*

• Four Fundamenatl elements

→ __Geometrical Instruments__

→ __Practical Geometry Fundamentals__

• Basic Shapes

→ __Copying Line and Circle__

• Basic Consustruction

→ __Construction of Perpendicular Bisector__

→ __Construction of Standard Angles__

→ __Construction of Triangles__

• Quadrilateral Forms

→ __Understanding Quadrilaterals__

→ __Construction of Quadrilaterals__

→ __Construction of Parallelograms__

→ __Construction of Rhombus__

→ __Construction of Trapezium__

→ __Construction of Kite__

→ __Construction of Rectangle__

→ __Construction of Square__