Polygon Angle Calculator

Polygon Angle Calculator for finding the interior, exterior and total angles of a regular polygon, then the miter, bevel and cut lengths to build it as a frame, a box or a planter.

Polygon Angle Calculator for Frames, Staves and Planters

The Polygon Angle Calculator gives the interior, exterior and total angles of any regular polygon, then turns them into saw settings. You can start from the number of sides, or from an angle you already know to find the shape.

It then sizes the shape from its width across flats, width across corners, side length, perimeter or area. Pick a flat frame, upright staves or a planter with sloped sides, and it returns the miter or bevel, cut lengths and a to-scale drawing.

Sizes are the outside of the finished shape, in inches or millimeters, and angles are in degrees. Enter actual milled stock sizes, such as 3/4 inch for a 1x board, since nominal sizes will throw off the short points and inside widths.

Interior, Exterior and Miter Angles by Side Count

These are exact geometry, not a shop rule of thumb. Each extra side adds one triangle of 180 degrees, and the turns around any closed polygon always add to 360 degrees.

$$S=(n-2)\times 180^{\circ }$$

$$A_{int}=\frac{(n-2)\times 180^{\circ }}{n}{A}_{ext}=\frac{360^{\circ }}{n}$$

A flat frame joint closes one exterior angle, and two pieces share it. So each cut is half the exterior angle, which is the number woodworkers actually set on the saw.

$$M=\frac{180^{\circ }}{n}$$

What angle do you cut for an octagon?

Set a miter saw to 22.5 degrees for each end of all eight pieces. That is half of the 45-degree exterior angle, not half of the 135-degree interior angle.

Some online charts list 67.5 degrees for the same frame. That figure is the complement, and it only applies to a gauge that reads 90 degrees at a square cut, as many table saw miter gauges do. The calculator shows both readings, so check which zero your saw uses before cutting.

Finding the side count from an angle

Start from an interior angle and the Polygon Angle Calculator works backward through the exterior angle. The same route works from an exterior angle or an angle sum.

$$n=\frac{360^{\circ }}{180^{\circ }-A_{int}}$$

A 150-degree interior angle gives 360 divided by 30, which is exactly 12 sides. Some homework answer sites give 13 for that same question, but the math leaves no remainder to round.

An angle that does not divide evenly has no regular polygon. A 100-degree interior angle works out to 4.5 sides, so the calculator stops and names the nearest shapes, a square at 90 degrees and a pentagon at 108.

Sizing a Hexagon or Octagon From Its Width

Most builds start from an overall width. For shapes with an even number of sides, the side length comes from the width across flats, and the width across corners follows from the side.

$$s=W_{flats}\times \tan \left(\frac{180^{\circ }}{n}\right)W_{corners}=\frac{s}{\sin (180^{\circ }/n)}$$

For an octagon, tan 22.5 degrees is 0.4142. A 24 inch octagon measured across flats has 9.941 inch sides and measures 25.977 inches point to point, with an area of 3.31 square feet.

A pentagon, heptagon or any shape with an odd side count has no parallel sides. Its width runs from one flat to the opposite corner, so the calculator relabels that field rather than treating it as across flats.

Short point and board length for a flat frame

The long point of each frame piece is the outside side length. The inside edge, or short point, is shorter by twice the frame width times the same tangent.

$$s_{short}=s-2w\tan \left(\frac{180^{\circ }}{n}\right)$$

A 2 inch wide octagon frame with 9.941 inch outside sides has 8.284 inch short points, and a 20 inch opening across flats. If the frame width eats the whole inside edge, the calculator stops and asks for a narrower frame or a larger shape.

Board length nests the pieces head to tail, with a saw kerf at every cut. The default 1/8 inch kerf lands the octagon at 74.812 inches of stock, before any allowance for knots or end checks.

Upright Staves and Sloped Planter Sides

When the sides stand upright, as in a staved column or a hexagon box, the part lies face down and the blade tilts instead. Each long edge gets a bevel of 180/n, the same number as the flat-frame miter.

A triangle needs a 60-degree bevel on each stave, which is past the tilt of most saws. Hexagons and octagons at 30 and 22.5 degrees sit well inside a normal tilt range, and the calculator also gives each stave's narrower inside face for the stock thickness you enter.

Sides that lean out, like a flared planter, need a compound cut. Festool's Kapex compound miter guide gives the two settings from the corner angle and slope. Here they are rewritten for a side tilt $T$ measured from vertical.

$$B=\arcsin \left(\cos T\times \sin \frac{180^{\circ }}{n}\right)$$

$$M_c=\arctan \left(\sin T\times \tan \frac{180^{\circ }}{n}\right)$$

An octagon planter with sides tilted 15 degrees needs a 21.69-degree bevel and a 6.12-degree miter. A square planter at 10 degrees needs a 44.14-degree bevel and a 9.85-degree miter.

Festool measures slope from the base, so a 15-degree lean is a 75-degree slope in its guide. Typing 75 into the tilt field swaps the geometry and gives settings for a nearly flat tray.

These settings assume the side lies face down on the saw table. Charts written for parts standing on edge against the fence swap the roles of miter and bevel, which is a common source of compound cuts that will not close.

Why Small Angle Errors Open the Last Joint

Every piece has two cuts, so an octagon has 16. On Woodwork Forums, a builder puts it plainly: the final joint of an octagon ends up out by 16 times the error on each cut.

A saw set 0.5 degrees off leaves an 8-degree gap at the last joint. On a 2 inch wide frame that opens 0.281 inch at the outer edge, far more than glue or clamps can hide.

That is why advice to "adjust by a degree" is too coarse for polygons. On an octagon, one degree per cut becomes 16 degrees at the closing joint, so the calculator also shows the gap at 0.1 degrees.

The same forum thread offers a practical fix. Glue the ring in two halves, then run the saw through both open joints at once, so the two halves meet on matching faces.

Wide miters also move with humidity, even when cut perfectly, because each face crosses the grain. Biscuits or splines across the joint help keep it aligned, and the biscuit joint calculator sizes and spaces them for the frame width.

Segmented turners use the same math, since each ring is a flat frame of short segments. Once the rings are glued up, the lathe speed calculator sets a safe speed for the finished blank's diameter.