Common angles and where they are used

The angles you meet again and again are 15, 22.5, 30, 45, 60, 90 and 120 degrees, and each has a structural reason. 45 halves a square corner, 22.5 halves it again and builds octagons, 30 and 60 come from the equilateral triangle, and 120 is the interior angle of a regular hexagon.

Publicado el 22 de julio de 2026 · Fuente: Angle properties of polygons, CIMT, University of Plymouth

Open a mitre saw and you will find detents cut into the base plate at 0, 15, 22.5, 31.6 and 45 degrees. Open a set of drafting triangles and you get 30, 45, 60 and 90. Look at a hex nut and you are looking at 120. These are not arbitrary conventions. They fall out of dividing a circle into a whole number of parts.

Learning why they recur makes them easier to use, because you can derive the one you need instead of hunting for a chart. This guide covers the source of each common angle, what it does in real work, and how to get it onto a workpiece without losing accuracy along the way.

Why do the same few angles keep appearing?

Because a circle has 360 degrees and 360 divides beautifully. It splits evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more, so equal divisions of a circle land on whole or half degrees. As the CIMT materials from the University of Plymouth put it, the exterior angles of any polygon always add up to 360 degrees, and each exterior angle of a regular polygon is 360 divided by the number of sides.

• Divide 360 by 3: exterior angle 120, interior angle 60. The equilateral triangle.
• Divide 360 by 4: exterior angle 90, interior angle 90. The square.
• Divide 360 by 6: exterior angle 60, interior angle 120. The hexagon.
• Divide 360 by 8: exterior angle 45, interior angle 135. The octagon.
• Divide 360 by 12: exterior angle 30, interior angle 150. The dodecagon, and the hour marks on a clock.

Halving does the rest. 90 halved is 45, halved again is 22.5, halved again is 11.25. Sixty halved is 30, halved again is 15. Almost every common angle in a workshop is either a direct division of the circle or one or two halvings away from one, which is also why folding paper reproduces so many of them exactly.

Why does 22.5 degrees matter so much?

Because it is the mitre angle for an octagon, and octagons are everywhere: gazebos, planters, clock faces, picture frames, drum shells, deck posts, stop signs. A regular octagon has an exterior angle of 45 degrees at each joint, and each of the two pieces meeting there takes half of it, which is 22.5 degrees.

The general rule is worth memorising, because it removes all the guesswork from frame building. For a flat frame with n equal sides, set the saw to 180 divided by n. A four sided frame is 180 divided by 4, or 45 degrees. A six sided frame is 30 degrees. An eight sided frame is 22.5 degrees. A three sided frame is 60 degrees.

• Triangle, 3 sides: mitre setting 60 degrees.
• Square or rectangle, 4 sides: mitre setting 45 degrees.
• Pentagon, 5 sides: mitre setting 36 degrees.
• Hexagon, 6 sides: mitre setting 30 degrees.
• Octagon, 8 sides: mitre setting 22.5 degrees.
• Twelve sided, 12 sides: mitre setting 15 degrees.

There is a second, quieter reason 22.5 gets its own detent on a mitre saw. Two cuts at 22.5 degrees, one on each of two pieces, produce a 45 degree change in direction, which is exactly what you need to turn a run of skirting board around a 45 degree wall corner. The detent is there because trim work asks for it constantly.

Accuracy compounds around a closed frame, which is why these numbers deserve care. If every cut on an eight sided frame is a tenth of a degree over, the error appears eight times in the same direction and the last joint is off by close to a degree, a visible gap. Cut one test joint, check it, adjust, and only then cut the real stock.

What angles are used for roof pitch?

Builders quote roof slope as rise over a 12 inch run rather than in degrees, so a 6/12 roof rises 6 inches for every 12 inches horizontally. Converting is a tangent calculation: 6 divided by 12 is 0.5, and the angle whose tangent is 0.5 is 26.57 degrees. The common pitches convert as follows.

• 2/12 pitch: 9.46 degrees. The minimum slope the International Residential Code allows for asphalt shingles, and only with double underlayment below 4/12.
• 3/12 pitch: 14.04 degrees.
• 4/12 pitch: 18.43 degrees. A very common minimum for standard shingle installation.
• 6/12 pitch: 26.57 degrees. The classic domestic roof.
• 8/12 pitch: 33.69 degrees.
• 10/12 pitch: 39.81 degrees.
• 12/12 pitch: 45 degrees, where rise equals run.

The reason two systems coexist is practical. On a roof you can lay a level and a tape and read rise over run directly, with no tool that measures degrees. On a drawing or a cutting list, degrees are what a saw scale understands. Roofing product literature tends to use one or the other and rarely both, so the conversion is a working necessity rather than a curiosity.

Where do 15, 30, 60 and 120 degrees turn up?

30 and 60 come from the equilateral triangle and run through drafting, quilting and metalwork. 120 is the interior angle of a hexagon, so it governs hex nuts, honeycomb patterns and hexagonal paving. 15 is the difference between the two standard drafting triangles, and it is the step size on many mitre saw detents.

• Quilting: 60 degree diamonds build hexagon and star blocks, and 45 degree cuts make half square triangles from squares.
• Drafting: the two standard set squares give 30-60-90 and 45-45-90, and stacking them yields 15 and 75 degrees.
• Metalwork: a standard twist drill has a 118 degree point angle, with 135 degrees used for harder materials and split point drills.
• Model making: 30 and 60 degree cuts dominate roof trusses and bracing at scale, because they come from equilateral geometry.
• Trim carpentry: 31.6 degrees appears as a mitre saw detent because it is the mitre setting for crown moulding at the common spring angle, paired with a bevel setting.
• Navigation and layout: 15 degrees is one hour of the earth rotating, which is why compass roses and sundials are marked in 15 degree steps.

A 45 degree angle deserves one specific warning. It is the only common angle where rise equals run, so people assume it is the easy one. In frames it is also the least forgiving, because two 45 degree cuts have to be complementary to within a fraction of a degree or the joint shows a wedge of light. If a mitre looks slightly open at the tip, the saw is off, not the wood.

How do I transfer an angle accurately onto a workpiece?

Mark the angle over the longest distance you can, not the shortest. Angular error is the same either way, but a line drawn 20 mm long amplifies it into a visible deviation over the length of the cut. Set the protractor centre exactly on the vertex, mark a point far out along the arc, then join with a straightedge.

1. Mark the vertex clearly with a knife nick or a fine pencil cross, not a fat dot. The centre of a 2 mm dot can be anywhere within a millimetre.
2. Lay the protractor baseline along the reference edge of the work, with the centre exactly on the vertex, and hold it down firmly.
3. Mark your degree with a single fine point as far out along the arc as the protractor allows. Further out means less angular error per unit of marking error.
4. Remove the protractor and join the vertex to the mark with a straightedge, extending the line right across the workpiece.
5. Check the line by measuring it back with the protractor turned over, reading from the other arm. The two readings should agree.
6. Cut a test piece in scrap first, offer the two halves together, and only then commit the real stock.

For angles that repeat many times, stop reading the protractor after the first one. Set a sliding bevel to the marked line, lock it, and use the bevel for every remaining piece. A single angle transferred mechanically twenty times is far more consistent than the same angle read from a scale twenty times, because each reading carries its own error.

The final check is the workpiece itself. For a 90 degree corner, use the 3-4-5 triangle over the largest distance available. For a mitred frame, dry assemble all the pieces before glue and look at the last joint, which accumulates every error in the set. If it closes, the angles were right.