What angle do I cut for an octagon?

Set your miter saw to 22.5° for all octagon joints. This is the miter angle: 180° ÷ 8 = 22.5°. Each joint brings two mitered ends together to form the 135° interior angle. Make sure your saw is accurately calibrated — even a 0.5° error multiplies across 8 joints and creates a 4° gap or overlap when you close the shape.

What is the miter angle for a hexagon?

The miter angle for a hexagon is 30° (180° ÷ 6 = 30°). Each interior angle is 120°. Hexagons are common in woodworking projects like planter boxes, decorative frames, and game boards. The circumradius of a regular hexagon equals the side length, which simplifies layout.

How do I calculate the area of a regular polygon?

Area = (number of sides × side length × apothem) ÷ 2. The apothem is the perpendicular distance from the center to the midpoint of any side: apothem = side length ÷ (2 × tan(π/n)). For example, an octagon with 48" sides has an apothem of 57.94" and an area of 11,086 sq in ≈ 77.0 sq ft.

What is the difference between circumradius and apothem?

The circumradius (R) is the distance from the center to any vertex (corner) — it’s the radius of the circle that passes through all corners. The apothem (r) is the distance from the center to the midpoint of any side — it’s the radius of the largest inscribed circle. The circumradius is always larger. For layout, use the circumradius to position corner posts and the apothem to position midpoint supports.

How do I lay out a polygon on the ground?

Drive a center stake, attach a string equal to the circumradius, and mark the first corner. Then use a string equal to the side length to step around from vertex to vertex. For higher accuracy, mark all vertices from the center using angles: divide 360° by the number of sides and use a protractor or transit to mark each radius. Check by measuring the diagonal — it should equal 2 × circumradius for even-sided polygons.

How many sides does a gazebo have?

Most gazebos are octagons (8 sides) or hexagons (6 sides). Octagonal gazebos are the most popular because the 135° interior angle creates a spacious, nearly circular interior while keeping the roof framing manageable. Hexagonal gazebos use 120° angles and are common for smaller structures. Square (4-sided) and dodecagonal (12-sided) gazebos are less common but perfectly buildable with the correct miter angle.

Why does my polygon not close perfectly?

Cumulative error is the most common cause. If your miter saw is off by even 0.5°, the total error across all joints is 0.5 × n (number of sides). For an octagon, that’s 4° of gap. Verify your saw angle with a digital angle finder, cut test pieces first, and dry-fit the polygon before final assembly. Also check that all sides are exactly the same length — even 1/16" variation per side accumulates to nearly 1/2" on an octagon.

Free Polygon Angle Calculator — Gazebo & Multi-Sided Angles

How to Use the Polygon Calculator

How to Calculate Polygon Angles and Dimensions

A regular polygon has all sides equal and all interior angles equal. The key formula for angles is: Interior Angle = (n − 2) × 180° ÷ n, where n is the number of sides. An octagon (8 sides) has interior angles of (8 − 2) × 180 ÷ 8 = 135°. The miter angle for your saw is half the exterior angle: 180° ÷ n. For an octagon, that’s 180 ÷ 8 = 22.5° — the exact setting for cutting gazebo, planter box, and picture frame miters.

Understanding Polygon Geometry

Every regular polygon has two important circles. The circumscribed circle (circumradius) passes through all vertices — its radius R determines the overall size. The inscribed circle (apothem or inradius) is the largest circle that fits inside the polygon, touching the midpoint of each side. The apothem is crucial for area calculations: Area = ½ × Perimeter × Apothem.

Circumradius: R = s ÷ (2 × sin(π/n)), where s is the side length
Apothem (inradius): r = s ÷ (2 × tan(π/n))
Area: A = (n × s × r) ÷ 2
Perimeter: P = n × s

Gazebo and Deck Applications

The most common polygon in residential construction is the octagon, used for gazebos, hot tub platforms, and decorative deck extensions. When building an octagonal gazebo with 48-inch sides, set your miter saw to 22.5° for all joint cuts. The circumradius tells you the distance from center to each corner post (about 62.6 inches for 48-inch sides), and the apothem gives you the distance from center to the midpoint of each side (about 57.9 inches). Use the apothem to position floor joists and the circumradius to lay out corner posts.

Pro Tips

Layout with a compass method: To lay out a polygon on the ground, drive a center stake, attach a string equal to the circumradius, and mark the first vertex. Then use a string equal to the side length to walk around, marking each successive vertex from the previous one.
Verify your miter saw: Before cutting all pieces, cut two test miters and check the joint. Even a 0.5° error in saw calibration multiplies around the polygon — an 8-sided shape accumulates 4° of total error from a 0.5° offset.
For large polygons: Work from the circumradius rather than the side length. Measure from center to each corner post, then connect them. This is more accurate than accumulating side-to-side measurements where errors compound.

Frequently Asked Questions

Compound Miter Calculator

Calculate miter and bevel angles for simple and compound cuts. Dual output modes for miter saw settings and protractor marking. Crown molding presets included.

Arc Calculator

Calculate arc dimensions from any two inputs: radius, chord, height, angle, or arc length. Stud layout for curved walls.

Feet & Inches Calculator

Add, subtract, multiply, and divide measurements in feet, inches, and fractions. Running tape, memory registers, and simultaneous imperial/metric output.

Tools for This Project

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Best Value

Swanson Tool S0101 Speed Square (7")

$12

Cast aluminum speed square for marking 90° and 45° cuts instantly. The degree scale along the hypotenuse lets you scribe any angle from 0° to 90° for custom polygon miters. Essential for layout and marking.

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Budget Pick

GemRed Digital Angle Finder (7")

$25

Digital angle gauge with 0.1° resolution for verifying polygon miter angles and checking saw calibration. Lock button holds the reading. Essential for ensuring your saw matches the calculated miter angle exactly.