Interactive Online Protractor
Drag the arms to measure any angle, or upload an image to measure angles on diagrams or photos.
- Acute: 0° < θ < 90°
- Right: θ = 90°
- Obtuse: 90° < θ < 180°
- Straight: θ = 180°
- Reflex: 180° < θ < 360°
Angle Classifications & Geometric Definitions
In Euclidean geometry, an angle is formed by two rays sharing a common endpoint called the vertex. Angles are categorized based on their magnitude relative to a straight line (180°) and a perpendicular right angle (90°):
Sharper than a square corner. Found in drafting triangles (30° and 60°), roof trusses, and arrowheads.
Formed by two perpendicular lines. The architectural foundation of walls, square frames, and Cartesian coordinate axes.
Wider than a right angle but less than a straight line. Typical in reclining chairs, octagon walls (135°), and wide bevels.
Forms a continuous flat line. The two rays point in exactly opposite directions from the central vertex point.
An angle greater than half a turn. Measured by switching the protractor into 360° Full circle mode.
One complete revolution where the measurement ray rotates through the entire plane and returns to the base ray.
Angular Measurement Systems: Degrees, Radians & Gradians
Depending on your field of work—such as civil engineering, computer graphics programming, physics, or woodworking—different angular units are standard:
| Unit System | Full Circle | Right Angle | Conversion to Degrees | Primary Use Case |
|---|---|---|---|---|
| Degrees (°) | 360° | 90° | 1° = (π / 180) rad | Everyday navigation, carpentry, architectural drafting. |
| Radians (rad) | 2π rad ≈ 6.283 | π / 2 ≈ 1.571 | 1 rad ≈ 57.2958° | SI scientific standard, calculus, trigonometry, physics. |
| Gradians (gon) | 400 gon | 100 gon | 1 gon = 0.9° | Surveying, European civil topographic engineering. |
Architectural Reference: Roof Pitch to Angle Conversion Table
In residential construction, roof slope is expressed as the number of vertical inches the roof rises for every 12 horizontal inches of run (e.g. 4/12 or 6/12 pitch). The angle in degrees is calculated using the inverse tangent function: θ = arctan(Rise / 12):
| Roof Pitch (Rise / 12) | Angle in Degrees | Angle in Radians | Slope Percentage | Common Construction Type |
|---|---|---|---|---|
| 2 / 12 Pitch | 9.46° | 0.165 rad | 16.7% | Low-slope roof, requires membrane roofing. |
| 3 / 12 Pitch | 14.04° | 0.245 rad | 25.0% | Minimum pitch for standard asphalt shingles. |
| 4 / 12 Pitch | 18.43° | 0.322 rad | 33.3% | Standard suburban ranch-style roof. |
| 6 / 12 Pitch | 26.57° | 0.464 rad | 50.0% | Most common North American residential roof. |
| 8 / 12 Pitch | 33.69° | 0.588 rad | 66.7% | Cape Cod style, excellent snow and rain runoff. |
| 10 / 12 Pitch | 39.81° | 0.695 rad | 83.3% | Tudor and Victorian architecture, steep attic space. |
| 12 / 12 Pitch | 45.00° | 0.785 rad | 100.0% | 45-degree gable roof, equal rise and run. |
Woodworking & Carpentry: Miter Saw Cuts for Multi-Sided Shapes
When building custom picture frames, polygonal planters, or decorative moldings, each joint consists of two angled miter cuts that meet to form the polygon's corner angle:
| Shape / Polygon | Number of Sides (n) | Interior Corner Angle | Miter Saw Blade Angle (180° / n) |
|---|---|---|---|
| Triangle | 3 Sides | 60.0° | 30.0° |
| Square / Rectangle | 4 Sides | 90.0° | 45.0° |
| Pentagon | 5 Sides | 108.0° | 36.0° |
| Hexagon | 6 Sides | 120.0° | 30.0° |
| Octagon | 8 Sides | 135.0° | 22.5° |
| Decagon | 10 Sides | 144.0° | 18.0° |
Frequently Asked Questions About Online Angle Measurement
How do I measure an angle on a physical object with this online protractor?▼
Place the corner or vertex of your object flat against your computer or tablet display. Align the object's bottom edge with the protractor's horizontal base ray (0° baseline). Then drag the blue handle of the moveable measurement arm until it coincides with your object's angled edge. The readout box instantly displays the exact angle in degrees, radians, and supplementary angles.
How do I measure angles in an uploaded photo or engineering schematic?▼
Click the 'Upload Image' button in the top toolbar and select any PNG, JPG, or WebP file (such as a floorplan, geometry assignment, architectural blueprint, or carpentry photo). The image will load directly behind the transparent protractor. Drag the center origin point and rotation arms over the image to measure angles directly on your diagram with zero server uploads.
What is the mathematical difference between degrees, radians, and gradians?▼
Degrees divide a full circle into 360 equal units based on ancient Babylonian astronomy. Radians are the SI metric unit for angular measurement, defined such that 1 radian produces an arc length equal to the circle's radius (a full circle is 2π radians ≈ 6.283 rad). Gradians (gon) divide a circle into 400 units, making a right angle exactly 100 gon. Instant Ruler computes live conversions between degrees and radians automatically.
How do I convert roof pitch (e.g. 4/12 or 6/12) to an angle in degrees?▼
Roof pitch expresses vertical rise over 12 inches of horizontal run. The formula to calculate degrees is: Angle = arctan(Rise / 12) × (180 / π). For example, a 4/12 pitch equals arctan(4/12) = 18.43°, while a 6/12 pitch equals arctan(6/12) = 26.57°, and a 12/12 pitch is exactly 45.00°.
What is the difference between supplementary and complementary angles?▼
Complementary angles sum to 90 degrees (forming a right angle). Supplementary angles sum to 180 degrees (forming a straight line). If an angle measures 40°, its complement is 50° (90° - 40°) and its supplement is 140° (180° - 40°). Our protractor provides the live supplementary angle readout in real time.
What miter and bevel angles do I use for cutting picture frames and polygons?▼
To calculate the miter saw cutting angle for any regular polygon, divide 180° by the number of sides (n): Miter Angle = 180° / n. For a standard 4-sided square picture frame, the cut is 180° / 4 = 45°. For a 6-sided hexagon box, the cut is 180° / 6 = 30°. For an 8-sided octagon, the cut is 180° / 8 = 22.5°.