A 45° angle is an acute angle: it is smaller than a right angle (90°). It is exactly half of a right angle, which makes it one of the most useful angles in geometry, carpentry and design.
45° is one eighth (1/8) of a full circle, or π/4 radians. If you face north and turn 45° clockwise, you end up facing due northeast (NE).
45° angle facts
| Angle type | Acute angle (between 0° and 90°) |
|---|---|
| Complement (adds to 90°) | 45° |
| Supplement (adds to 180°) | 135° |
| Explement (adds to 360°) | 315° |
| Radians | π/4 ≈ 0.785398 |
| Gradians | 50 grad |
| Fraction of a turn | one eighth (1/8) (12.5%) |
| Degrees, minutes, seconds | 45° 0′ 0″ |
| Slope (grade) | 100% |
| Roof pitch | 12/12 |
| Compass bearing | northeast (NE) |
Trigonometric values of 45°
| Function | Value |
|---|---|
| sin 45° | √2/2 ≈ 0.707107 |
| cos 45° | √2/2 ≈ 0.707107 |
| tan 45° | 1 |
45° on a protractor
How to draw a 45° angle with a protractor
- Draw a straight line with a ruler. This is the first arm; mark the vertex at one end.
- Place the center mark of the protractor exactly on the vertex and line up the 0° baseline with your line.
- Find 45° 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.
- Remove the protractor and draw a line from the vertex through the dot. Mark the angle with an arc and label it 45°.
Drawing it a 45° angle without a protractor (compass and ruler)
- Construct a 90° angle, then bisect it.
- Paper method: fold a square piece of paper corner to corner. The fold makes a 45° angle with each side.
Where you see 45° angles
- The diagonal of a square cuts its corners into two 45° angles.
- Picture frames and door casings are joined with 45° miters.
- A 12/12 roof pitch rises 12 inches for every 12 inches of run, a 45° roof.
- Plumbing "45" elbows turn a pipe by 45°.
- Divide a circle into 8 equal slices and each slice is 45°, e.g. 8 spokes on a wheel or 8 equal pie-chart wedges.
- A board sloped at 45° rises about 120 inches over a 10-foot (120-inch) run.
- 45° of latitude on Earth spans roughly 3,107 miles north to south.
Related angles
45° angle FAQ
What type of angle is 45°?
A 45° angle is an acute angle, because it is between 0° and 90°.
What are the complement and supplement of 45°?
The complement of 45° is 45° (together they make 90°), and the supplement is 135° (together they make 180°).
How many radians is 45 degrees?
45° = π/4 radians ≈ 0.7854 rad. To convert degrees to radians, multiply by π/180: 45 × π/180 = π/4.
What is 45° as a slope or roof pitch?
45° is a 100% slope, which is a roof pitch of 12/12 (a rise of about 12 inches for every 12 inches of run).