How to Construct a 75° Angle: Step-by-Step Instructions and Examples

Constructing a 75° angle is an important geometry skill that helps students understand how to construct angles accurately using simple tools like a ruler and compass. Mastering this concept strengthens the foundation of geometric construction and develops precision in mathematical drawing. A 75° angle is commonly constructed by combining basic angles, making it an excellent exercise for improving logical thinking and problem-solving abilities. It is useful for school exams, assignments and practical geometry work. In this guide, you'll learn a simple, step-by-step method for accurately constructing a 75° angle using standard geometric techniques.

Table of Contents

• How to Construct a 75° Angle
• Why the 75° Angle Construction Works
• Common Mistakes to Avoid During 75° Angle Construction
• Frequently Asked Questions of How to Construct a 75 Degree Angle

How to Construct a 75° Angles

Materials Required to Construct a 75° Angle: 

• A pencil

• A ruler or straightedge (for drawing straight lines only)

• A pair of compasses that can hold its width steady

Constructing a 75° Angle Using a Compass and Ruler

Stage 1 · Build the 60° ray

Step 1: Draw a ray starting at point O passing through point A.

Step 2: With O as centre and a convenient radius, draw a large arc cutting ray OA at B.

Step 3: Without changing the radius, place the compass at B and draw an arc cutting the first arc at C. Join O to C, the ray OC makes a 60° angle with OA.

Stage 2 · Extend to the 90° ray

Step  4: Keeping the same radius, place the compass at C and draw another arc cutting the first big arc at D. Join O to D, the  ray OD makes a 120° angle with OA.

Step 5:  With C and D as centres and a radius more than half the distance CD, draw two arcs above so they intersect at point E. Join O to E and extend it to meet the first arc at M. Ray OE bisects ∠COD, so it makes exactly a 90° angle with OA (since 60° + 30° = 90°).

Stage 3 · Bisect between 60° and 90°

Step 6: With C and M as centres and a radius more than half the distance CM, draw two arcs so they intersect at point H. Join O to H. Ray OH bisects the angle between OC (60°) and OE (90°).

Step 7: The angle ∠AOQ is exactly 75°.

Why the 75° Angle Construction Works

• Triangle OBC has all three sides equal (same compass radius), making it equilateral, so ∠BOC = 60°.

• Triangle OCD is equilateral by the same logic, so ∠COD = 60°, which makes ∠BOD = 60° + 60° = 120°.

• Bisecting ∠COD splits that 60° gap into two 30° halves, so ray OP sits at 60° + 30° = 90° from the base.

• Bisecting the new gap between OC (60°) and OP (90°) splits that 30° difference into two 15° halves, placing ray OQ at 60° + 15° = 75°.

(60° + 90°) ÷ 2 = 150° ÷ 2 = 75°

Common Mistakes to Avoid During 75° Angle Construction

• Bisecting the wrong angle

The final bisection must happen between the 60° ray and the 90° ray; not between the 60° ray and the 120° ray and not between the base ray and the 90° ray. Bisecting the wrong pair gives 90°, 60° or 45° instead of 75°.

• Changing compass width across stages

The radius used to construct 60° and 120° doesn't need to match the radius used for the later bisections, but it must stay fixed within each individual stage. Switching mid-arc breaks the equal-radius property the whole method depends on.

• Skipping the 90° ray entirely

Some students try to shortcut straight from 60° to 75° by guessing a bisection. Without first fixing the 90° ray accurately, there's nothing correct to bisect against.

• Losing track of labelled points

With four or five rays fanning out from the same vertex, unlabelled arcs quickly become impossible to tell apart. Label every intersection the moment you draw it.