Constructing Angles
Using a compass and ruler, you can construct certain angles without a protractor. The key angles you can construct are 60°, 90°, 120°, 30°, 45°, and others derived from these.
You can also copy any given angle using a compass — without knowing its measurement.
In Class 6, you will learn step-by-step constructions for these standard angles.
What is Constructing Angles - Grade 6 Maths (Practical Geometry)?
Angles you can construct with compass and ruler:
- 60° — the base construction (uses equilateral triangle property)
- 120° — double of 60°
- 90° — perpendicular construction
- 30° — bisect 60°
- 45° — bisect 90°
- 150° — 180° − 30°
Copying an angle: You can copy any angle from one location to another using a compass, even without knowing its measurement.
Constructing Angles Formula
Construction of 60° angle:
- Draw a ray OA.
- With O as centre and any radius, draw an arc cutting OA at point P.
- With P as centre and the same radius, draw an arc cutting the first arc at point Q.
- Join OQ and extend it.
- ∠AOQ = 60°.
Construction of 120° angle:
- Draw a ray OA. With O as centre, draw an arc cutting OA at P.
- With P as centre and same radius, cut the arc at Q (this is 60°).
- With Q as centre and same radius, cut the arc again at R (this is 120°).
- Join OR.
- ∠AOR = 120°.
Construction of 90° angle:
- Construct 60° (getting point Q) and 120° (getting point R) as above.
- With Q and R as centres, draw arcs with equal radius to meet at S.
- Join OS.
- ∠AOS = 90° (the bisector of the 60°-to-120° region).
Construction of 30° — bisect 60°:
- Construct a 60° angle (getting OA and OQ).
- Bisect ∠AOQ: from P and Q draw arcs with equal radius → meet at T.
- Join OT.
- ∠AOT = 30°.
Types and Properties
Copying an angle (using compass):
- You are given angle ∠ABC. You want to copy it at a new point.
- Draw a ray DE.
- With B as centre, draw an arc cutting both rays of ∠ABC at P and Q.
- With D as centre and the same radius, draw an arc cutting DE at P'.
- Measure the distance PQ with the compass. With P' as centre and radius = PQ, draw an arc cutting the first arc at Q'.
- Join DQ' and extend.
- ∠EDQ' = ∠ABC.
Constructing other angles:
- 45° = bisect 90°
- 150° = construct 30° from the straight angle side (180° − 30°)
- 135° = 90° + 45° (or 180° − 45°)
- 75° = 60° + bisect the remaining 30° (i.e., bisect 60° to 90°)
Solved Examples
Example 1: Constructing 60°
Problem: Construct an angle of 60° at point O.
Solution:
- Draw ray OA.
- Arc from O cuts OA at P.
- Arc from P (same radius) cuts previous arc at Q.
- Join OQ.
Answer: ∠AOQ = 60°.
Example 2: Constructing 120°
Problem: Construct 120° at point O.
Solution:
- Draw ray OA. Arc from O cuts OA at P.
- From P, cut arc at Q (60°). From Q, cut arc at R (120°).
- Join OR.
Answer: ∠AOR = 120°.
Example 3: Constructing 30°
Problem: Construct an angle of 30°.
Solution:
- First construct 60° (getting ray OA and ray OQ).
- Bisect the 60° angle: arcs from P and Q meet at T.
- Join OT.
Answer: ∠AOT = 30°.
Example 4: Constructing 90°
Problem: Construct a 90° angle.
Solution:
- Draw ray OA. Arc from O gives P on OA.
- From P, mark Q (60°). From Q, mark R (120°).
- Bisect ∠QOR: arcs from Q and R meet at S.
- Join OS.
Answer: ∠AOS = 90°.
Example 5: Constructing 45°
Problem: Construct a 45° angle.
Solution:
- First construct 90° (getting ray OA and ray OS).
- Bisect the 90° angle.
Answer: The bisector makes 45° with OA.
Example 6: Copying an Angle
Problem: Copy angle ∠PQR to a new ray DE.
Solution:
- Arc from Q cuts rays QP and QR at A and B.
- Arc from D (same radius) cuts DE at A'.
- Measure AB with compass. From A', mark B' on the arc.
- Join DB'.
Answer: ∠EDB' = ∠PQR.
Example 7: Constructing 150°
Problem: Construct an angle of 150°.
Solution:
- Draw a straight line (180°). Mark point O.
- Construct 30° from one side of the straight line.
- The remaining angle = 180° − 30° = 150°.
Answer: The angle on the other side = 150°.
Real-World Applications
Where angle constructions are used:
- Engineering: Machine parts need exact angle specifications.
- Architecture: Roof slopes, staircase angles, and window frames.
- Navigation: Setting direction at a specific angle from north.
- Art: Creating geometric patterns like stars and mandalas.
- Further geometry: Constructing triangles and polygons requires angle construction.
Key Points to Remember
- 60° is the base construction — it uses the property that an equilateral triangle has 60° angles.
- 120° = two 60° arcs from the same starting point.
- 90° = bisect the angle between 60° and 120°.
- 30° = bisect 60°. 45° = bisect 90°.
- To copy an angle, use a compass to transfer the arc and the opening.
- Always use the same compass radius when marking arcs on the base arc.
- Do not erase construction arcs.
Practice Problems
- Construct an angle of 60° using compass and ruler.
- Construct an angle of 30° by bisecting 60°.
- Construct an angle of 90°.
- Construct an angle of 45° by bisecting 90°.
- Construct an angle of 120°.
- Copy a given angle from one location to another using compass.
Frequently Asked Questions
Q1. Why can I construct 60° easily with a compass?
When you draw an arc from a point, and then use the same radius to cut the arc from the point where it meets the ray, you create an equilateral triangle setup. Each angle of an equilateral triangle is 60°.
Q2. Can I construct any angle with compass and ruler?
No. Only certain angles (like 60°, 90°, 120°, 30°, 45°, and their multiples and halves) can be constructed exactly. Other angles need a protractor.
Q3. How do I construct 75°?
Construct 60° and 90°. Then bisect the angle between them. The bisector gives 75°.
Q4. What if my compass slips?
The construction will be inaccurate. Tighten the compass screw and hold it firmly at the top while rotating gently.
Q5. Do I need to measure the final angle with a protractor?
Not for the construction itself, but you can use a protractor to verify your construction is accurate.
Related Topics
- Constructing Perpendicular Lines
- Bisecting an Angle
- Types of Angles
- Measuring Angles with Protractor
- Constructing a Line Segment
- Bisecting a Line Segment
- Constructing Parallel Lines
- Constructing Triangles (SSS)
- Constructing Triangles (SAS)
- Constructing Triangles (ASA)
- Constructing Triangles (RHS)
- Constructing Quadrilaterals
- Constructing Special Quadrilaterals
- Triangle Construction Problems
