40 Degree Angle

At 40°, this angle is acute, meaning it is narrower than the corner of a square. It is less than half of a right angle, and its complement (the angle that makes 90° with it) is 50°.

40° is one ninth (1/9) of a full circle, or 2π/9 radians. If you face north and turn 40° clockwise, you end up facing between NNE and NE (closest to northeast (NE)).

40° angle facts

Trigonometric values of 40°

40° on a protractor

How to draw a 40° angle with a protractor

1. Draw a straight line with a ruler. This is the first arm; mark the vertex at one end.
2. Place the center mark of the protractor exactly on the vertex and line up the 0° baseline with your line.
3. Find 40° on the row of numbers that reads 0° on your line (protractors have two rows running in opposite directions) and make a small dot at the edge.
4. Remove the protractor and draw a line from the vertex through the dot. Mark the angle with an arc and label it 40°.

Drawing it a 40° angle without a protractor (rise and run method)

1. Draw a straight baseline and measure exactly 10 inches along it from the vertex.
2. At the 10-inch mark, draw a short line straight up at 90° (use a carpenter's square or the corner of a sheet of paper).
3. Measure 8.39 inches (about 8 3/8″) up that line and make a mark.
4. Join the mark to the vertex. The angle at the vertex is 40°, because tan 40° ≈ 0.8391.

Where you see 40° angles

• Divide a circle into 9 equal slices and each slice is 40°, e.g. 9 spokes on a wheel or 9 equal pie-chart wedges.
• A 10/12 roof pitch is 39.81°, very close to 40°.
• A board sloped at 40° rises about 100.7 inches over a 10-foot (120-inch) run.
• 40° of latitude on Earth spans roughly 2,762 miles north to south.
• The Earth turns 40° on its axis in about 2 h 40 min.

Related angles

• Complement: 50°
• Supplement: 140°
• Explement: 320°
• Half: 20°
• Double: 80°
• 30°
• 45°
• 60°
• 90°