In the "Angles of an n-gon" mode, enter the number of angles (vertices) n. In the "Regular polygon from one angle" mode, enter the size of one angle of the regular polygon.
Enter numbers only. The number of angles (vertices) n is a whole number from 3 to 360. For angle sizes, decimals and fractions such as 900/7 are OK.
Result and figure
Enter the number of angles (vertices) in the field on the left and press "Calculate". The result and a figure of the regular polygon will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the number of angles (vertices) of an \(n\)-gon (3 to 360), and you get the sum of the interior angles \(180^\circ \times (n - 2)\), the sum of the exterior angles \(360^\circ\), and each interior and exterior angle of a regular \(n\)-gon on the spot
- Angles that do not divide evenly, such as each interior angle of a regular heptagon, \(\dfrac{900}{7}^\circ\), are shown both as an exact fraction and as a decimal. The steps also show the idea of splitting the polygon into \((n-2)\) triangles
- It also works backward: from the size of one interior (or exterior) angle, it finds how many sides the regular polygon has. If the number of vertices is not a whole number, it tells you that no regular polygon has that angle
- The result includes a figure of the regular polygon, so you can see where one interior angle and one exterior angle are and how the diagonals split it into triangles (with too many angles, the figure or the diagonals are left out)
The explanation of exterior angles on this page assumes a polygon with no dents (a convex polygon).
What is this calculation used for?
Checking survey errors (land surveying and mapping)
When surveyors measure property lines, they treat the lot as a polygon and measure the angle at each corner. By checking how far the total of the measured interior angles is from the theoretical value \(180^\circ \times (n - 2)\) (the angular misclosure), they can catch measuring mistakes and instrument errors.
This check is a standard daily step in real survey work, and it is an example of the interior angle sum formula being used directly for quality control in the field.
Tiling floors and the shape of honeycombs (design and nature)
To cover a floor with one kind of regular polygon and no gaps, the interior angles must fit exactly around one point (\(360^\circ\)). So only shapes whose interior angle divides \(360^\circ\) evenly can tile the plane: the equilateral triangle (\(60^\circ\)), the square (\(90^\circ\)) and the regular hexagon (\(120^\circ\)). Regular pentagons (\(108^\circ\)) always leave gaps.
Honeycombs are hexagons because this shape divides space with no gaps while using little material.
Drawing shapes with code (coding classes and robots)
To draw a regular polygon in Scratch, with turtle graphics or with a robot, you repeat "move forward a little, then turn a little". The turning angle is not the interior angle but the exterior angle (\(360^\circ \div n\)). For a regular pentagon, turning \(72^\circ\) each time brings you back to the start and closes the shape.
This is a classic lesson in school coding classes, and it turns the idea "the exterior angles add up to \(360^\circ\) (one full turn)" directly into a program.
Camera apertures and the shape of light (photography and optics)
A camera lens controls the amount of light with an aperture, an opening close to a regular polygon formed by several blades. The number of blades (the number of angles of the polygon) changes how many rays stream out from lights in a night photo and the shape of the background blur.
In product design, the number of angles of the polygon and its angles are part of what gives each lens its own look.
Formulas and figures
Sum of the interior angles of an \(n\)-gon
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(180^\circ\)
\(\times\)
\((n - 2)\)
In words
(symbols replaced with words)
③
\(S\): sum of the interior angles of the \(n\)-gon
\(=\)
②
\(180^\circ\): angle sum of a triangle
\(\times\)
①
\(n - 2\): number of triangles
The formula in words
①
Cut the \(n\)-gon with diagonals from one vertex and take the
\(n - 2\): number of triangles
②
multiply it by the
\(180^\circ\): angle sum of a triangle
③
and you get the
\(S\): sum of the interior angles of the \(n\)-gon
Quick example
A pentagon (\(n = 5\)) can be split by diagonals into \(5 - 2 = 3\) triangles, so the sum of its interior angles is
sum of the interior angles of a pentagon
\(=\)
angle sum of a triangle (\(180^\circ\))
\(\times\)
number of triangles (3)
\(180^\circ \times (5 - 2) = 180^\circ \times 3 = 540^\circ\)
Key idea
Why \(n - 2\)? From one vertex of an \(n\)-gon, you can draw diagonals to \(n - 3\) vertices (all except itself and its two neighbors). These \(n - 3\) diagonals cut the polygon into \(n - 2\) triangles, and all the angles of those triangles together make up exactly the interior angles of the polygon. So the sum is "\(n - 2\) times the triangle angle sum of \(180^\circ\)".
Check it with a quadrilateral: one diagonal splits it into 2 triangles, so the sum is \(180^\circ \times 2 = 360^\circ\). A triangle needs no splitting, and \(180^\circ \times 1 = 180^\circ\) matches the value you already know.
Sum of the exterior angles of a polygon
Figure
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(T\)
\(=\)
\(360^\circ\)
①
\(T\): sum of the exterior angles
\(=\)
②
\(360^\circ\): exactly one full turn
The formula in words
①
For any polygon, the
\(T\): sum of the exterior angles
②
is
\(360^\circ\): exactly one full turn
Quick example
Checking with a pentagon - there are 5 interior-exterior pairs of \(180^\circ\) each, so subtract the interior sum of \(540^\circ\)
sum of the exterior angles of a pentagon
\(=\)
\(360^\circ\) (exactly one full turn)
\(180^\circ \times 5 - 540^\circ = 900^\circ - 540^\circ = 360^\circ\)
Key idea
Here is an intuitive reason why the exterior angles of any polygon add up to \(360^\circ\). Imagine walking once around the polygon along its sides. At each vertex, you turn to face the next side, and the angle you turn is exactly the exterior angle. When you are back at the start, you face the same way as when you began, so you have turned exactly one full turn (\(360^\circ\)) in total. That is why all the exterior angles always add up to \(360^\circ\).
You can also check it by calculation. At each vertex, interior + exterior \(= 180^\circ\) (a straight angle), so all \(n\) vertices together give \(180^\circ \times n\). Subtract the interior sum \(180^\circ \times (n - 2)\), and what is left is always \(180^\circ \times 2 = 360^\circ\). The more angles there are, the smaller each exterior angle becomes, but the total stays the same.
This holds for polygons with no dents (convex polygons).
Each interior angle of a regular \(n\)-gon
Standard notation
(the usual math form)
\(x\)
\(=\)
\(180^\circ \times (n - 2)\)
\(\div\)
\(n\)
In words
(symbols replaced with words)
③
\(x\): each interior angle of the regular \(n\)-gon
\(=\)
①
\(180^\circ \times (n - 2)\): sum of the interior angles
\(\div\)
②
\(n\): number of angles
The formula in words
①
All interior angles of a regular polygon are equal, so take the
\(180^\circ \times (n - 2)\): sum of the interior angles
②
divide it evenly by the
\(n\): number of angles
③
and you get
\(x\): each interior angle of the regular \(n\)-gon
Quick example
Each interior angle of a regular pentagon (\(n = 5\)) is the interior sum of \(540^\circ\) divided into 5 equal parts
each interior angle of a regular pentagon
\(=\)
sum of the interior angles (\(540^\circ\))
\(\div\)
number of angles (5)
\(180^\circ \times (5 - 2) \div 5 = 540^\circ \div 5 = 108^\circ\)
Key idea
You can divide evenly like this only for a regular polygon. For an ordinary pentagon, only the total of \(540^\circ\) is fixed; the individual interior angles can all be different.
The division does not always come out even. For a regular heptagon, \(900^\circ \div 7 = \dfrac{900}{7}^\circ = 128.57\ldots^\circ\), which is not a whole number. On a test, it is safest to give the answer as an exact fraction such as \(\dfrac{900}{7}^\circ\).
You can also find it as "\(180^\circ -\) one exterior angle" (using interior + exterior \(= 180^\circ\)). Finding the exterior angle first as \(360^\circ \div n\) with the next formula and then subtracting is often easier.
Each exterior angle of a regular \(n\)-gon (and finding \(n\) from an angle)
Standard notation
(the usual math form)
\(y\)
\(=\)
\(360^\circ\)
\(\div\)
\(n\)
In words
(symbols replaced with words)
③
\(y\): each exterior angle of the regular \(n\)-gon
\(=\)
①
\(360^\circ\): sum of the exterior angles
\(\div\)
②
\(n\): number of angles
The formula in words
①
All exterior angles of a regular polygon are equal too, so take the
\(360^\circ\): sum of the exterior angles
②
divide it evenly by the
\(n\): number of angles
③
and you get
\(y\): each exterior angle of the regular \(n\)-gon
Quick example
Each exterior angle of a regular pentagon (\(n = 5\)), and going backward, "if each exterior angle is \(72^\circ\), how many sides?"
each exterior angle of a regular pentagon
\(=\)
sum of the exterior angles (\(360^\circ\))
\(\div\)
number of angles (5)
\(360^\circ \div 5 = 72^\circ\)
\(360^\circ \div 72^\circ = 5\)
Key idea
This formula also works backward. If you know the size \(y\) of one exterior angle, the number of angles is \(n = 360^\circ \div y\). For example, an exterior angle of \(40^\circ\) gives \(360^\circ \div 40^\circ = 9\), a regular nonagon.
If it does not divide evenly, no regular polygon has an exterior angle of that size (as with \(360^\circ \div 50^\circ = 7.2\), the number of vertices is not a whole number).
To work back from one interior angle \(x\), first turn it into the exterior angle \(y = 180^\circ - x\) using interior + exterior \(= 180^\circ\), then calculate \(360^\circ \div y\). The backward mode of this calculator follows the same steps.
The sum of the interior angles of an \(n\)-gon is \(180^\circ \times (n - 2)\), because diagonals from one vertex split it into \(n - 2\) triangles. The sum of the exterior angles of any polygon is exactly one full turn, \(360^\circ\). In a regular \(n\)-gon all angles are equal, so each interior angle is the interior sum \(\div\ n\) and each exterior angle is \(360^\circ \div n\). Going backward, the size of one angle tells you how many sides the regular polygon has.
Symbols and terms
Symbols
| \(n\) | n | The number of angles (vertices). For a pentagon, \(n = 5\). The letter \(n\), from "number", is often used for a count. |
| \(S\) | S | The letter for the sum of the interior angles on this page, from the first letter of "sum". It is also often used for totals and for area. |
| \(T\) | T | The letter for the sum of the exterior angles on this page, from the first letter of "total". |
| \(x,\ y\) | x, y | Letters for the angles you want to find. Textbooks usually call an unknown number \(x\), as in "let each interior angle be \(x^\circ\)". |
| \(^\circ\) | degree | The symbol for the degree, the unit of angle. One full turn divided into 360 equal parts is \(1^\circ\). The number 360 is said to have been chosen because it has many factors and is easy to divide evenly. |
Terms
| polygon | A closed figure made of three or more line segments (straight sides). It is the general name for triangles, quadrilaterals, pentagons and so on. A polygon with \(n\) angles is called an \(n\)-gon. |
| vertex | A corner point of a polygon, where two sides meet (plural - vertices). An \(n\)-gon has \(n\) of them. The numbers of angles, vertices and sides are all the same. |
| interior angle | An angle on the inside of a polygon. An \(n\)-gon has \(n\) of them, the same as its number of vertices. |
| exterior angle | The angle between one side of a polygon extended as a line and the next side. The interior and exterior angles at the same vertex together make a straight angle (\(180^\circ\)). |
| sum of the interior angles | All the interior angles added together. For an \(n\)-gon it is \(180^\circ \times (n - 2)\). A triangle has \(180^\circ\), a quadrilateral \(360^\circ\) and a pentagon \(540^\circ\); each extra angle adds \(180^\circ\). |
| sum of the exterior angles | One exterior angle taken at each vertex, all added together. For a polygon with no dents, it is always exactly one full turn, \(360^\circ\), whatever the number of sides. |
| regular polygon | A polygon whose sides are all the same length and whose interior angles are all the same size, such as an equilateral triangle, a square or a regular pentagon. Because all the angles are equal, "interior sum ÷ number of angles" gives each interior angle. |
| diagonal | A line segment joining two vertices of a polygon that are not next to each other. From one vertex you can draw \(n - 3\) of them (to every vertex except itself and its two neighbors), and they cut the polygon into \(n - 2\) triangles. |
| triangle angle sum | The fact that the three interior angles of any triangle always add up to \(180^\circ\). The formula for the interior angle sum of a polygon comes from splitting the polygon into triangles and using this fact. |
| straight angle | The angle of a straight line. It measures \(180^\circ\), exactly half a turn. "Interior + exterior \(= 180^\circ\)" comes from the fact that a side and its extension form a straight line, that is, a straight angle. |
| convex polygon | A polygon with no dents, that is, every interior angle is less than \(180^\circ\). The explanation of exterior angles on this page (exterior angle sum \(= 360^\circ\)) assumes a convex polygon. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to the topics in this list is the quickest way forward.
| Angle sum of a triangle (Grades 5–8) |
|
| Angles and turns (Grade 4) |
|
| Polygons and diagonals (Grades 3–7) |
|
| Division and fractions (Grades 5–6) |
|
| One-variable equations (Grades 6–7) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the sum of the interior angles of an n-gon
| Number of angles (vertices) n | 5 |
| Sum of interior angles S | =180*(B1-2) |
Table to find each interior angle of a regular n-gon
| Number of angles (vertices) n | 5 |
| Each interior angle x | =180*(B1-2)/B1 |
Table to find each exterior angle of a regular n-gon
| Number of angles (vertices) n | 5 |
| Each exterior angle y | =360/B1 |
Table to find the regular polygon from one interior angle
| Each interior angle (degrees) | 108 |
| Each exterior angle (degrees) | =180-B1 |
| Number of vertices n = 360 ÷ exterior angle | =360/B2 |
After pasting, the upper rows are your inputs and the lower rows are calculated automatically.
"*" is multiplication and "/" is division.
The first table finds the sum of the interior angles of a pentagon (n = 5), and the answer is 540. The second and third tables give each interior angle and each exterior angle of a regular pentagon, 108 and 72.
The fourth table works backward. Enter the interior angle 108, and you get an exterior angle of 72 and 5 vertices, so it is a regular pentagon. If the number of vertices is not a whole number, such as 7.2, no regular polygon has that angle.
"*" is multiplication and "/" is division.
The first table finds the sum of the interior angles of a pentagon (n = 5), and the answer is 540. The second and third tables give each interior angle and each exterior angle of a regular pentagon, 108 and 72.
The fourth table works backward. Enter the interior angle 108, and you get an exterior angle of 72 and 5 vertices, so it is a regular pentagon. If the number of vertices is not a whole number, such as 7.2, no regular polygon has that angle.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the sum of the interior angles of an n-gon
| Number of angles (vertices) n | 5 |
| Sum of interior angles S | =180*(B1-2) |
Table to find each interior angle of a regular n-gon
| Number of angles (vertices) n | 5 |
| Each interior angle x | =180*(B1-2)/B1 |
Table to find each exterior angle of a regular n-gon
| Number of angles (vertices) n | 5 |
| Each exterior angle y | =360/B1 |
Table to find the regular polygon from one interior angle
| Each interior angle (degrees) | 108 |
| Each exterior angle (degrees) | =180-B1 |
| Number of vertices n = 360 ÷ exterior angle | =360/B2 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace n or the angle with your own numbers.
How to calculate it in Python
from fractions import Fraction
# Angles of an n-gon (vertex_count = number of angles/vertices)
vertex_count = 7
interior_sum = 180 * (vertex_count - 2)
regular_interior = Fraction(interior_sum, vertex_count)
regular_exterior = Fraction(360, vertex_count)
print(f"Sum of interior angles: {interior_sum}°")
print("Sum of exterior angles: 360°")
print(f"Each interior angle of a regular {vertex_count}-gon: {regular_interior}° ≈ {float(regular_interior):.2f}°")
print(f"Each exterior angle of a regular {vertex_count}-gon: {regular_exterior}° ≈ {float(regular_exterior):.2f}°")
# Backward: find the regular polygon from one interior angle
interior_angle = Fraction(150)
exterior_angle = 180 - interior_angle
polygon_count = Fraction(360) / exterior_angle
if polygon_count.denominator == 1 and polygon_count >= 3:
print(f"Regular polygon with an interior angle of {interior_angle}°: regular {polygon_count}-gon")
else:
print(f"No regular polygon has an interior angle of {interior_angle}° (360 ÷ exterior angle = {polygon_count})")
With the fractions module from the standard library, angles that do not divide evenly, such as 900/7°, stay exact fractions with no rounding error. The first half of this example is for a regular heptagon (vertex_count = 7), and each interior angle is shown as "900/7° ≈ 128.57°". The second half works back from an interior angle of 150° to an exterior angle of 30° and 12 vertices, and shows "regular 12-gon". Change the numbers and run it.
How to write it in LaTeX and other math languages (copy and paste)
Sum of the interior angles of an \(n\)-gon
S = 180° × (n − 2)
S = 180^{\circ} \times (n - 2)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<msup><mn>180</mn><mo>°</mo></msup>
<mo>×</mo>
<mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo>
</mrow>
</math>
S = 180^circ xx (n - 2)
S = 180 (n - 2) Degree
S := 180*(n - 2);
S = 180*(n - 2);
S = 180° × (n − 2)
Sum of the exterior angles of a polygon
T = 360°
T = 360^{\circ}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>T</mi>
<mo>=</mo>
<msup><mn>360</mn><mo>°</mo></msup>
</mrow>
</math>
T = 360^circ
T = 360 Degree
T := 360;
T = 360;
T = 360°
Each interior angle of a regular \(n\)-gon
x = 180° × (n − 2) / n
x = \frac{180^{\circ} \times (n - 2)}{n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow><msup><mn>180</mn><mo>°</mo></msup><mo>×</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow>
<mi>n</mi>
</mfrac>
</mrow>
</math>
x = (180^circ xx (n - 2))/n
x = (180 (n - 2)/n) Degree
x := 180*(n - 2)/n;
x = 180*(n - 2)/n;
x = 180° × (n − 2) / n
Each exterior angle of a regular \(n\)-gon (and finding \(n\) from an angle)
y = 360° / n
y = \frac{360^{\circ}}{n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<msup><mn>360</mn><mo>°</mo></msup>
<mi>n</mi>
</mfrac>
</mrow>
</math>
y = (360^circ)/n
y = (360/n) Degree
y := 360/n;
y = 360/n;
y = 360° / n
How to have ChatGPT do the calculation
You are a math (plane geometry) calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For a regular heptagon, show each of the following: 1. The sum of the interior angles (degrees) 2. The size of each interior angle (both as a fraction in lowest terms and as a decimal) 3. The size of each exterior angle (both as a fraction in lowest terms and as a decimal) Also find how many sides a regular polygon with an interior angle of 140° has, by first converting it to the exterior angle and then dividing 360° by it. In Python, use the fractions module from the standard library to calculate exactly, and show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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