GCSE Maths Geometry and Measure Constructions

How to Construct a 30, 60, 45, 90 Degree Angle

# How to Construct a 30, 60, 45, 90 Degree Angle

Here we will learn about how to construct a 30, 60, 45 and 90 degree angle using a pencil, a ruler and a pair of compasses.

There are also constructions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What do we mean by constructing a 30, 60, 45, 90 degree angle?

Constructing a 30, 60, 45, 90 degree angle is constructing these angles accurately without using a protractor.

To do this we need to use a pencil, a ruler (a straight-edge) and compasses.

E.g.

A 60 degree angle can be constructed by drawing an equilateral triangle.
Then an angle bisector will construct a 30 degree angle.

E.g.

A 90 degree angle can be constructed with a perpendicular bisector.
Then an angle bisector will construct a 45 degree angle.

### What is constructing a 30, 60, 45, 90 degree angle? ## How to construct a 60 degree angle

In order to construct a 60 degree angle:

1. Draw line.
2. From one end of the line draw an arc.
3. From where the arc crosses the line, draw another arc.
4. Join the end point of the line to where the two arcs intersect.

### How to construct a 60 degree angle ## How to construct a 60 degree angle example

### Example 1: construct a 60 degree angle

Construct a 60° angle

1. Draw a line.

With a pencil and a straight-edge draw a straight line

2From one end of the line draw an arc.

Use compasses centered on one end of the line, draw an arc.

3From where the arc crosses the line, draw another arc.

Keep the compasses at the same setting. The new arc should cross the first arc.

4Join the end point of the line to where the two arcs intersect.

Use a straight-edge to join the end point used in step 2 to the intersection of the two arcs.

## How to construct a 30 degree angle

In order to construct a 30 degree angle:

1. Construct a 60 degree angle.
2. Construct an angle bisector of the 60 degree angle.

### How to construct a 30 degree angle ## How to construct a 30 degree angle example

### Example 2: construct a 30 degree angle

Construct a 30° angle

Follow the steps in Example 1 to construct a 60 degree angle

Angle ABC is 60°

Place the point of your compasses at A and draw an arc. Keep the compasses at the same setting and repeat placing the point at C and drawing another arc. Using a straight-edge, join up the point where the arcs intersect each other with the vertex B .

The 60 degree angle has been bisected into two equal angles, both 30 degrees.

## How to construct a 90 degree angle

In order to construct a 90 degree angle:

1. Draw a line.
2. Construct a perpendicular bisector.

### How to construct a 90 degree angle ## How to construct a 90 degree angle example

### Example 3: construct a 90 degree angle

Construct a 90° angle

Set your compasses to about three-quarters of the length of your line. Place the point of your compasses on one endpoint of the line and draw an arc. Keeping the compasses set at the same setting, draw another arc from the other endpoint. Finally use a straight-edge to join the two intersections of the arcs.

The new line is the perpendicular bisector of the original line segment.

## How to construct a 45 degree angle

In order to construct a 45 degree angle:

1. Construct a 90 degree angle.
2. Construct an angle bisector of the 90 degree angle.

### How to construct a 45 degree angle ## How to construct a 45 degree angle example

### Example 4: construct a 45 degree angle

Construct a 45° angle

Follow the steps in Example 3 to construct a 90 degree angle

Place the point of your compasses at the centre A and draw an arc. Place the point of your compasses at B and draw an arc. Keep the compasses at the same setting and repeat.  Place the compasses at C and draw another arc. Using a straight-edge, join up the point where the arcs intersect each other with the centre A .

The 90 degree angle has been bisected into two equal angles, both 45 degrees.

### Common misconceptions

• The pencil should be sharp

A sharp pencil helps your diagram to be accurate. Using a small pencil in compasses can also be helpful.

How to construct a 30, 60, 45, 90 degree angle is part of our series of lessons to support revision on constructions. You may find it helpful to start with the main constructions lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

### Practice how to construct a 30, 60, 45, 90 degree angle questions

1. Construct a 60 degree angle        The construction must be made with compasses kept to the same setting for both arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.

2. Construct a 30 degree angle        The construction must be made with compasses kept to the same setting for both the last arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.

3. Construct a 90 degree angle        The construction must be made with compasses kept to the same setting for both the last arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.The construction must be made with compasses kept to the same setting for both the last arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.

4. Construct a 45 degree angle        The construction must be made with compasses kept to the same setting for both the last arcs. The construction arcs must be seen. The lines should be drawn with a straight-edge.

### How to construct a 30, 60, 45, 90 degree angle GCSE questions

1. In the space below, using compasses and a ruler, construct a 90^{\circ} angle.

(2 marks) For drawing the construction arcs

(1)

For completing the perpendicular bisector

(1)

2. In the space below, using compasses and a ruler, construct a 30^{\circ} angle.

(3 marks) For drawing the construction arcs to make a 60 degree angle

(1)

For drawing the arcs to construct an angle bisector

(1)

For completing the construction of the 30 degree angle

(1)

## Learning checklist

You have now learned how to:

• Construct a 60 degree angle
• Construct a 30 degree angle
• Construct a 90 degree angle
• Construct a 45 degree angle

## Still stuck?

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#### FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR.

Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.