Last updated: July 15, 2026
Polygon Angle Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The Polygon Angle Calculator solves regular polygon angle relationships in two directions. It uses Interior angle = (n−2)·180/n and Exterior angle = 360/n, and it can also recover the side count with n = 360/(180 − interior angle) when that inverse value corresponds to a whole-number regular polygon.
For a regular polygon, divide 360 by the number of sides to get the exterior angle, subtract from 180 for the interior angle, or reverse the process to recover the side count from a valid interior angle.
Key Takeaways
- Each exterior angle of a regular polygon equals 360 ÷ n.
- Each interior angle equals (n − 2) × 180 ÷ n.
- Interior and exterior angles at one vertex always add to 180 degrees.
- Inverse mode is valid only when the computed side count is essentially a whole number.
- Benchmark angles such as 108°, 120°, 135°, 140°, and 150° make fast checks easier.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Interior angle = (n−2)·180/n; Exterior angle = 360/n; n = 360/(180 − interior angle)
Where:
- n=Number of sides
- \theta_i=Interior angle(degrees)
- \theta_e=Exterior angle(degrees)
- S=Sum of interior angles(degrees)
Worked Examples
From sides: regular 12-gon
Start with the side count when you want the angle measures directly.
- 1Use n = 12 in the regular polygon angle formulas.
- 2Compute the interior angle as (12 − 2) × 180 ÷ 12 = 150°.
- 3Compute the exterior angle as 360 ÷ 12 = 30°.
- 4Compute the interior-angle sum as (12 − 2) × 180 = 1800°.
From interior angle: 150 degrees
Inverse mode identifies the polygon when the interior angle matches a whole-number regular shape.
- 1Start with n = 360 ÷ (180 − 150).
- 2The denominator is 30, so n = 360 ÷ 30 = 12.
- 3Use n = 12 to compute the exterior angle as 30°.
- 4Use n = 12 to compute the total interior-angle sum as 1800°.
From interior angle: 140 degrees
This example shows a nonagon recovered from its interior angle.
- 1Start with n = 360 ÷ (180 − 140).
- 2The denominator is 40, so n = 360 ÷ 40 = 9.
- 3The exterior angle is therefore 360 ÷ 9 = 40°.
- 4The full interior-angle sum is (9 − 2) × 180 = 1260°.
Introduction
The Polygon Angle Calculator focuses on the fixed angle structure of regular polygons. In fromSides mode, it turns a whole-number side count into the interior angle, exterior angle, and sum of interior angles. In fromInteriorAngle mode, it reverses the relationship to recover the side count when the angle truly belongs to a regular polygon. That makes the tool useful for geometry homework, sketch validation, and fast checks of whether an angle measurement can describe a whole-number regular polygon at all.
The basic angle relationships in a regular polygon
Regular polygon angles are tightly linked because symmetry forces every corner and every turning step to behave the same way. If a polygon has n sides, its interior angles together add to (n − 2) × 180 degrees. Because the polygon is regular, each interior angle is that total divided by n. The exterior angle relationship is even more direct: moving around the polygon once creates one full 360-degree turn, so each exterior angle must be 360 ÷ n. Those two values are supplementary, which means a regular polygon always satisfies interior angle plus exterior angle equals 180 degrees. This calculator packages those linked facts into two modes. In one mode, you start with the number of sides and compute the angles. In the other, you start with a valid interior angle and check whether it belongs to a regular polygon with a whole-number side count. Understanding the geometry behind those modes makes the output much easier to trust and explain.
Sum of interior angles is (n − 2) × 180 degrees.
Each exterior angle equals 360 ÷ n.
Interior and exterior angles add to 180 degrees.
Regularity is what makes one corner stand for every corner.
Using the fromSides mode
The fromSides mode is the direct path when you already know how many sides the regular polygon has. Enter an integer n of at least 3, and the calculator immediately returns the interior angle, the exterior angle, and the full sum of the interior angles. This is the mode most often used in geometry class because many textbook problems begin with a named polygon such as a pentagon, hexagon, or dodecagon. It is also the best mode for quick reference work, since the answer depends only on the side count and not on any edge length. A helpful mental check is to notice that larger n values produce larger interior angles and smaller exterior angles. A triangle has a wide exterior turn of 120 degrees, while a 12-gon turns only 30 degrees at each vertex. If your result violates that trend, the input side count is probably wrong or non-regular assumptions have crept into the problem.
Use a whole-number side count n ≥ 3.
Expect interior angles to grow as n increases.
Expect exterior angles to shrink as n increases.
Use named polygons as quick side-count benchmarks.
Using the fromInteriorAngle mode
The fromInteriorAngle mode works backward from the angle to the polygon. Starting with a regular polygon interior angle θ, the side count is n = 360 ÷ (180 − θ). That formula comes from the supplementary relationship between the interior and exterior angles. Once the exterior angle is 180 − θ, the number of equal exterior turns that fit into 360 degrees gives the side count. This mode is especially useful when a problem states that each interior angle has a certain measure and asks you to identify the polygon. The crucial detail is that the computed n must land on a whole number. An interior angle of 150 degrees gives n = 12, so it corresponds to a regular dodecagon. An interior angle of 100 degrees gives n = 4.5, which cannot describe a regular polygon with a whole-number side count. That is why the calculator validates the inverse result instead of blindly rounding it.
Work backward with n = 360 ÷ (180 − interior angle).
Interior angles must be strictly between 0 and 180 degrees.
A valid regular polygon requires a whole-number side count.
Inverse mode is ideal for identification problems.
Why the whole-number check matters
Not every angle between 0 and 180 degrees belongs to a regular polygon. The inverse formula may produce a positive number, but that number still has to represent an actual count of sides. This is why the calculator checks whether the computed n is within a tiny tolerance of a whole number rather than just rounding casually. Without that safeguard, an invalid angle could be turned into a misleading polygon label. For example, 140 degrees is valid because it leads to exactly 9 sides. By contrast, 100 degrees gives 4.5 sides, which has no geometric meaning in this context. The distinction matters in proofs, exam problems, and software pipelines where clean validation is just as important as the arithmetic itself. Treat the whole-number test as part of the mathematics, not as a programming technicality. It captures the fact that regular polygons come in discrete families, not a continuous slider of side counts.
A positive inverse result is not automatically valid.
The side count must represent a real whole polygon.
Tolerance prevents floating-point noise from blocking valid cases.
Validation protects against false identifications.
Reading the outputs across both modes
The calculator always reports the number of sides, the interior angle, the exterior angle, the sum of the interior angles, and the mode used. In fromSides mode, the side count mainly acts as the starting point and the angle values become the new information. In fromInteriorAngle mode, the angle is the starting point and the side count becomes the main discovery. The mode output is useful because it makes the calculation path explicit, which helps when results are copied into notes, homework, or application code. The sum of the interior angles gives a whole-shape perspective and can be checked independently once n is known. The exterior angle is often the quickest sanity check because it must divide evenly into 360 degrees for a valid regular polygon. If you read the outputs as a connected system instead of isolated numbers, it becomes much easier to catch impossible values before they cause confusion downstream.
Use mode to remember which direction the problem was solved.
Treat exterior angle as a quick divisibility check of 360 degrees.
Use the total interior-angle sum as a whole-shape verification step.
Read side count and angle values as a linked geometry system.
Common regular polygons and their angle measures
Many classroom questions revolve around a small set of named polygons, so it helps to know their angle patterns without recomputing them every time. The table below shows the most common cases that appear in geometry courses, drafting sketches, and quick mental checks. Notice the steady progression. A triangle has the smallest interior angle in the regular family, while a dodecagon already looks much closer to a circle and therefore has a much larger interior angle. At the same time, the exterior angle shrinks because each step around the figure turns less sharply. Memorizing every entry is unnecessary, but recognizing these anchor values makes it much easier to estimate answers and spot impossible ones. For instance, if someone claims that a regular hexagon has an exterior angle of 45 degrees, the table immediately shows that 45 degrees belongs to an octagon instead.
Named polygons are useful mental anchors.
Interior angles rise as polygons gain sides.
Exterior angles fall as polygons gain sides.
Reference values make error detection much faster.
| Polygon | Sides n | Interior angle (°) | Exterior angle (°) |
|---|---|---|---|
| Triangle | 3 | 60 | 120 |
| Square | 4 | 90 | 90 |
| Pentagon | 5 | 108 | 72 |
| Hexagon | 6 | 120 | 60 |
| Octagon | 8 | 135 | 45 |
| Nonagon | 9 | 140 | 40 |
| Decagon | 10 | 144 | 36 |
| Dodecagon | 12 | 150 | 30 |
Interior angles that map cleanly to whole-number polygons
When you use inverse mode repeatedly, a few interior angles appear again and again. These are the angles that convert neatly into whole-number side counts, making them especially useful for study and for quick mental recognition. The pattern also reveals why some angles fail validation. Every valid interior angle in this context must leave an exterior angle that divides 360 evenly. That is the real structural test hiding underneath the inverse formula. Looking at the table, you can see that 150 degrees implies a 30-degree exterior angle and therefore 12 equal turns, while 140 degrees implies a 40-degree exterior angle and therefore 9 equal turns. By contrast, an interior angle such as 100 degrees leaves an 80-degree exterior angle, and 360 divided by 80 is not an integer. Studying the inverse table is one of the fastest ways to turn the formulas into intuition.
Valid interior angles leave exterior angles that divide 360 evenly.
Inverse mode is really a divisibility test in disguise.
Benchmark angle values help identify polygons quickly.
Non-matching angles are invalid even if they look reasonable.
| Interior angle (°) | Exterior angle (°) | Number of sides n | Regular polygon |
|---|---|---|---|
| 90 | 90 | 4 | Square |
| 108 | 72 | 5 | Pentagon |
| 120 | 60 | 6 | Hexagon |
| 135 | 45 | 8 | Octagon |
| 140 | 40 | 9 | Nonagon |
| 144 | 36 | 10 | Decagon |
| 150 | 30 | 12 | Dodecagon |
Common mistakes in polygon angle work
Students often mix up the total sum of the interior angles with the measure of each interior angle. The total uses (n − 2) × 180, but a regular polygon’s single interior angle requires dividing that total by n. Another mistake is forgetting that interior and exterior angles are supplementary, which leads to inverse formulas written with the wrong subtraction. In inverse problems, some people round the side count even when the calculation does not land close to a whole number. That produces neat-looking but invalid answers. It is also common to forget that the input angle must be strictly between 0 and 180 degrees for a convex regular polygon. Finally, learners sometimes assume that angle information alone works for irregular polygons, but equal-angle and equal-side symmetry is what makes these formulas reliable. A slow check of the definitions usually fixes the issue long before the arithmetic becomes a problem.
Do not confuse total interior angle sum with one interior angle.
Keep the supplementary relationship interior + exterior = 180° in view.
Never round a clearly non-integer side count into a fake polygon.
Remember that these are regular-polygon formulas, not general polygon formulas.
Where polygon angle relationships are useful
Polygon angle calculations show up in geometry classes, but they also matter in design, graphics, and layout work. Whenever a repeating shape must fit evenly around a center or along a rotation, the exterior angle becomes the practical control value. Interior angles matter in tiling studies, drafting constraints, and any problem where corner openness affects construction. Inverse angle work is common in exam questions because it tests whether students understand the structure rather than just memorizing a direct formula. Software that generates regular shapes also depends on the same relationships, since vertex placement and turning steps are driven by angle rules. Even if you never draw a polygon by hand, the formulas teach an important habit: connect a geometric statement, an algebraic expression, and a validation rule before accepting an answer. That blend of reasoning and checking is exactly what this calculator is designed to support.
Use exterior angles to control even rotational steps.
Use interior angles to reason about corner geometry.
Use inverse mode to identify polygons from angle data.
Use the relationships as checks in graphics and design workflows.
Quick Reference Card
Polygon Angle Calculator Cheat Sheet
Quick reference • Polygon Angle Calculator
Interior angle = (n−2)·180/n; exterior angle = 360/n; inverse side count = 360 ÷ (180 − interior angle)Valid range: Use an integer n ≥ 3 in fromSides mode, or use an interior angle strictly between 0 and 180 degrees in fromInteriorAngle mode.
Common Values
⚠ Watch Out
- •Do not use inverse mode for angles outside the open interval from 0° to 180°.
- •A computed side count must be a whole number to represent a regular polygon.
- •Do not confuse the sum of all interior angles with one interior angle.
- •These formulas apply only to regular polygons.
Pro Tips
- →Check that 360 is divisible by the exterior angle.
- →Use named polygons as anchors when estimating the answer.
- →Verify that interior angle plus exterior angle equals 180°.
- →Treat inverse mode as both a formula step and a validity test.
FAQs
Why do interior and exterior angles add to 180 degrees?
At each vertex of a regular polygon, the interior angle and the exterior turning angle form a straight line pair. Straight angles measure 180 degrees, so the two values are always supplementary.
Why is the side count required to be an integer?
A polygon is defined by a whole number of straight sides. Fractional values such as 4.5 do not describe actual polygons, so the calculator rejects them.
Can any angle between 0 and 180 degrees be the interior angle of a regular polygon?
No. The angle must also produce a whole-number side count through n = 360 ÷ (180 − interior angle). If that inverse calculation does not land on a whole number, the angle is invalid for a regular polygon.
Why does a larger polygon have a smaller exterior angle?
The full turn around any polygon is 360 degrees. When more equal sides share that turn, each exterior turning step becomes smaller.
What does the sum of interior angles tell me?
It gives the total angle measure across the entire polygon, not the size of one corner. It is useful as a whole-shape check once the side count is known.
Why is 150 degrees valid but 100 degrees invalid in inverse mode?
An interior angle of 150 degrees gives n = 12, which is a whole number. An interior angle of 100 degrees gives n = 4.5, so it does not correspond to any whole-number regular polygon.
Does this calculator work for irregular polygons?
No. The formulas depend on every side and every interior angle being equal. Irregular polygons need different information and different methods.