What is the area formula for a regular polygon?

For a regular polygon with n equal sides of length s, the area is (1/4)·n·s²·cot(pi/n). The cotangent term encodes how the corners open up as n grows. Worked example, a regular hexagon with s=1, area = (1/4)·6·1·cot(30 degrees) = 1.5·1.7320508 = 2.598076. A square (n=4, s=2) gives (1/4)·4·4·cot(45 degrees) = 4·1 = 4, which matches s² as expected.

How do I find the interior angle of a regular polygon?

Each interior angle equals (n-2)·180/n degrees, because the polygon splits into n-2 triangles whose angles total (n-2)·180, shared equally across n corners. A triangle (n=3) gives 60 degrees, a square 90, a pentagon 108, and a regular hexagon (n-2)·180/6 = 4·180/6 = 120 degrees. As n grows the interior angle climbs toward 180 and the shape rounds out toward a circle.

What is the exterior angle and how does it relate to the interior one?

The exterior angle is 360/n degrees, the turn you make at each corner walking once around the outline. Every exterior angle pairs with its interior angle to sum to 180, so interior + exterior = 180 always. For a square 360/4 = 90, for a hexagon 360/6 = 60, for a decagon (n=10) 360/10 = 36. The exterior angles of any convex polygon always add up to a full 360 degrees turn.

What is the apothem and how is it different from the circumradius?

The apothem is the distance from the center to the midpoint of a side, also called the inradius, the radius of the inscribed circle that touches every side. It equals s / (2·tan(pi/n)). The circumradius reaches from the center to a vertex, the radius of the circle through all corners, and equals s / (2·sin(pi/n)). The circumradius is always the larger of the two because a vertex sits farther out than a side midpoint.

What is the sum of the interior angles of a polygon?

The interior angles of any simple n-sided polygon add up to (n-2)·180 degrees, regular or not. A triangle totals 180, a quadrilateral 360, a pentagon 540, a hexagon 720. For a regular polygon you divide that sum by n to get one angle. The exterior angles tell a tidier story, they always sum to 360 no matter how many sides, which is why a single full lap around the shape turns you through one complete circle.

Does this work for shapes with many sides?

Yes, enter any whole number of sides from 3 upward. With n=100 you get a near-circle, interior angle 176.4 degrees, and the area approaches that of the circumscribed circle. The named shapes cover triangle through dodecagon (n=12), and anything past 12 is labelled as an n-gon, for example a 20-gon. The math stays exact double-precision throughout, so a chiliagon with n=1000 still computes cleanly even though it looks like a circle on screen.

Check geometry homework before handing it in

A worksheet asks for the area and interior angle of a regular octagon with 5 cm sides. Enter n=8 and s=5, read area = 120.71 cm² and interior angle = 135 degrees, then compare against your own working. The live drawing confirms you pictured an octagon and not a hexagon, and the apothem and circumradius are right there if the next part of the question needs them.

Lay out a gazebo, tabletop or planter base

Building a hexagonal deck or an octagonal planter, you know the side length you want from the lumber on hand. Punch in n and s to get the apothem (how far the center sits from each edge) and the circumradius (how far out the corners reach), the two numbers you need to mark the layout on the ground and check it clears the space before any cuts.

Teach the (n-2)·180/n pattern in class

Step a class through triangle, square, pentagon, hexagon and watch the interior angle climb 60, 90, 108, 120 while the exterior angle falls 120, 90, 72, 60. The shareable URL lets you drop a specific shape into a lesson slide or assignment so every student opens the exact same figure rather than a blank tool.

Size a bolt circle or CNC pocket pattern

A flange needs six holes on a regular hexagon. The circumradius is your bolt-circle radius and the central angle between holes is the exterior angle, 360/6 = 60 degrees. Read both off one screen, copy the result into your notes, and reproduce the layout later from the link without re-deriving anything.

Regular Polygon Calculator, Area, Interior Angle and Apothem

Sides n + side length s, get area, interior angle, apothem and both radii, with a to-scale n-gon drawn live, browser-only

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Number of sides (n)

Side length (s)

Shape: Hexagon

Area: 2.598076
Perimeter: 6
Interior angle: 120°
Exterior angle: 60°
Sum of interior angles: 720°
Apothem (inradius): 0.866025
Circumradius: 1

Regular polygon (to scale)

— circumradius-- apothem

What this tool does

A regular polygon calculator that takes the number of sides n and one side length s and returns every standard measurement at once: perimeter ns, area (1/4)·n·s²·cot(pi/n), the interior angle (n-2)·180/n, the exterior angle 360/n, the sum of interior angles, the apothem (the inradius, distance from center to the midpoint of a side), and the circumradius (distance from center to a vertex). It names the shape too, so n=6 reads back as a hexagon and n=12 as a dodecagon. A live SVG draws the actual polygon to scale with its apothem and circumradius marked, so you see the geometry rather than only the numbers. Every formula runs as plain JavaScript in your tab, one click copies the full result, and the share link reproduces your exact n and s. 100% client-side, no upload.

Tool details

Input: Files + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy + Preview; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 10 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Regular Polygon Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Check geometry homework before handing it in
A worksheet asks for the area and interior angle of a regular octagon with 5 cm sides. Enter n=8 and s=5, read area = 120.71 cm² and interior angle = 135 degrees, then compare against your own working. The live drawing confirms you pictured an octagon and not a hexagon, and the apothem and circumradius are right there if the next part of the question needs them.
Lay out a gazebo, tabletop or planter base
Building a hexagonal deck or an octagonal planter, you know the side length you want from the lumber on hand. Punch in n and s to get the apothem (how far the center sits from each edge) and the circumradius (how far out the corners reach), the two numbers you need to mark the layout on the ground and check it clears the space before any cuts.
Teach the (n-2)·180/n pattern in class
Step a class through triangle, square, pentagon, hexagon and watch the interior angle climb 60, 90, 108, 120 while the exterior angle falls 120, 90, 72, 60. The shareable URL lets you drop a specific shape into a lesson slide or assignment so every student opens the exact same figure rather than a blank tool.
Size a bolt circle or CNC pocket pattern
A flange needs six holes on a regular hexagon. The circumradius is your bolt-circle radius and the central angle between holes is the exterior angle, 360/6 = 60 degrees. Read both off one screen, copy the result into your notes, and reproduce the layout later from the link without re-deriving anything.

Common pitfalls

Mixing up the apothem and the circumradius. The apothem reaches the midpoint of a side (inradius), the circumradius reaches a vertex. The circumradius is always larger. Using the apothem where a corner-to-center distance was needed undersizes the shape.
Using degrees inside the formulas. The area and radius formulas use cot, tan and sin of pi/n in radians, not 180/n in degrees. Feeding degrees into a calculator set to radians (or the reverse) is the most common source of a wrong area.
Entering fewer than 3 sides. A polygon needs at least 3 sides, n=1 or n=2 is not a closed shape and the tool refuses it. Also remember n must be a whole number, there is no regular 5.5-gon.

Privacy

Every result here, the area, the angles, the apothem, the circumradius and the polygon drawing, is computed by plain JavaScript inside your browser tab. Nothing about your shape is sent anywhere and nothing is logged. The one thing to know, the shareable link encodes n and s in the URL query string, so a link pasted into chat will record those two numbers in the recipient server access log. The values are harmless geometry, but if you would rather not leave any trace, use the copy button and paste the text instead of sharing the URL.

FAQ

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Regular Polygon Calculator, Area, Interior Angle and Apothem

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Regular Polygon Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Check geometry homework before handing it in

Lay out a gazebo, tabletop or planter base

Teach the (n-2)·180/n pattern in class

Size a bolt circle or CNC pocket pattern

Common pitfalls

Privacy

FAQ

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