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Free Online Polygon Calculator, Regular and Irregular

Work out the area, perimeter, interior angle, apothem and number of diagonals of a regular polygon from its number of sides and side length, or the area and perimeter of any irregular polygon from the coordinates of its corners. The coordinate method works whichever way round the corners are listed.

Free Forever Nothing Uploaded Regular And Irregular Either Direction

Free Online Polygon Calculator, Regular and Irregular

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⬡ Polygon Calculator Regular from sides · irregular from coordinates · angles and diagonals

The sum runs in your browser. Nothing you type is sent anywhere.

What you have

⬡ The polygon

A regular polygon has every side the same length and every angle the same size, so the number of sides and one length describe it completely.

Number of sides

Length of one side

Try one of these

The last two are the same rectangle with the corners listed in opposite directions, and they give the same area.

The area

—

Sides—

Perimeter—

Each interior angle—

All the interior angles—

Apothem—

Radius to a corner—

Diagonals—

Recent polygons, kept in this browser only.

Nothing here yet Press Log to history on the Calculate tab to keep a polygon here.

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How to Find the Area of a Polygon

Three steps, and the second depends on what you have.

Say whether the polygon is regular A regular one has every side and angle the same, and the number of sides with one length describes it completely. An irregular one has no such shortcut and needs its corners.

Enter what you have For a regular polygon, the number of sides and a side length. For an irregular one, the coordinates of the corners in order around the outline, one per line. Either direction round works.

Read the area and the rest A regular polygon also gives its interior angle, apothem, radius to a corner and number of diagonals. An irregular one gives the area, the perimeter and which way the corners were listed.

What to Know About Polygons

Regular against irregular, and what each one needs.

Listing the corners the other way round gives the same area. The shoelace formula produces a signed value, and its sign flips when the corners are listed clockwise instead of anticlockwise. An area is never negative, so the sign is reported separately as the direction you listed them in, and the area itself is always positive. Calculators that hand back a negative area have simply forgotten to take the absolute value.

Three sides is the minimum, and the formula does not know it. The apothem formula happily returns a number for two sides, and two sides enclose nothing at all. The check that there are at least three comes before the formula rather than being left to it, because a formula producing an answer is not evidence that the shape exists.

More sides means closer to a circle. A regular polygon with a hundred sides has an area within a hundredth of a percent of the circle through its corners, and its interior angle is 176.4 degrees. That convergence is how Archimedes computed pi, and it is visible here by raising the number of sides and watching the apothem close on the radius.

The interior angles always add to the same total. For any polygon with n sides, regular or not, the interior angles add to 180 times n minus two degrees. A quadrilateral comes to 360, a pentagon to 540. Only in a regular polygon is each angle that total divided by n.

The diagonal count grows quickly. Each corner joins to every other except itself and its two neighbours, and each diagonal is counted twice that way, so the total is n times n minus three, over two. A pentagon has five, a decagon has thirty five, and a twenty sided polygon has a hundred and seventy.

It runs in your browser with nothing uploaded. The arithmetic is in the page, so it works offline once loaded and nothing you type is sent anywhere.

Key Features & Capabilities

What it does, and what it deliberately does not.

Regular and irregular From the number of sides and a length, or from the coordinates of any set of corners.

Direction does not matter Corners listed either way round give the same positive area, with the direction reported separately.

Angles, apothem and radius Each interior angle, the total, the distance to a side and the distance to a corner.

Diagonals counted Because the formula is easy to get wrong by forgetting that each one is counted twice.

The working shown Including the shoelace sum term by term for an irregular polygon.

Nothing is uploaded It runs in this tab and keeps working offline.

About Polygons

A polygon is a closed shape made of straight sides. When every side and every angle is the same it is called regular, and that symmetry is powerful: the number of sides and a single length describe the shape completely, and everything else, the area, the angles, the distance to a side and the distance to a corner, follows from those two numbers.

An irregular polygon has no such shortcut. Its area depends on exactly where each corner sits, so the input has to be the coordinates themselves. The method for turning them into an area is the shoelace formula, which walks around the outline multiplying each x by the next y and subtracting each y times the next x. The name comes from how the terms cross over when written out in two columns.

The shoelace formula has one property that confuses people, which is that its result is signed. Walking around the corners anticlockwise gives a positive value and clockwise gives a negative one of the same size. An area is not a negative quantity, so this page always reports it as positive and states the direction separately. The sign is genuinely useful information, since it tells you how your corner list is ordered, but it is not part of the area.

Regular polygons are worth playing with because of what happens as the number of sides increases. The apothem, which is the distance from the centre to the middle of a side, creeps up towards the radius out to a corner. The interior angle creeps up towards 180 degrees. The area creeps up towards the area of the circle. With a hundred sides the polygon and its circle agree to within a hundredth of a percent, which is essentially how Archimedes pinned down pi more than two thousand years ago.

Frequently Asked Questions

Sides, angles, coordinates and the shoelace formula.

Every side the same length and every angle the same size. A square is regular; a general rectangle is not. Only regular polygons can be described by a number of sides and a single length.

List the corners one per line as an x and a y, going round the outline in order. Either direction works, and there is no need to repeat the first corner at the end.

The direction does not change the area, only the sign of the intermediate calculation, which is reported separately. The order does matter in the sense that the corners must go round the outline rather than jumping about, or the shape being described is a different one.

The distance from the centre of a regular polygon to the midpoint of a side, at right angles to it. It is the radius of the largest circle that fits inside, and the area is half the perimeter times the apothem.

Because two straight sides cannot enclose any space. The formulas will happily return a number for two, which is why the check is made before they run rather than being left to them.

n times n minus three, all over two. Each corner joins to every other except itself and its two neighbours, and dividing by two accounts for each diagonal being counted from both ends.

No. The arithmetic runs in your browser, so nothing you type leaves the page and it works offline.