Geometry

Regular polygons and their angles

When every side and every angle of a polygon is the same, how big is each angle, and which sizes are possible at all?

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When you reach for it

A regular polygon is a polygon whose sides all have the same length and whose interior angles all have the same size. A polygon with all its sides equal is called equilateral, and one with all its angles equal is called equiangular. Regular means both at once.

Both conditions are needed, because from four sides on neither one forces the other.

Figure 1Equal sides without equal angles, equal angles without equal sides, and both at once. Only the hexagon on the right is regular.

The rhombus in Figure 1 has four equal sides, but two of its angles are 60^\circ and two are 120^\circ. The rectangle has four equal angles, all 90^\circ, but its long sides are not the same length as its short ones. Only the hexagon has both, so only the hexagon is regular.

Triangles are the one exception. A triangle with three equal sides has three equal angles, and the other way round, by the facts about isosceles and equilateral triangles. So a regular triangle is just an equilateral triangle, and a regular quadrilateral is a square.

Reach for this lesson when you know a polygon is regular, or only that its angles are all equal, and you want the size of one angle. Reach for it the other way round when you know one angle and want the number of sides, or want to know whether a regular polygon with that angle exists at all. Both rest on two facts proved in angle sums of polygons: the interior angles of an n-gon add to (n - 2) \cdot 180^\circ, and the exterior angles, one at each vertex of a convex polygon, add to 360^\circ. This lesson shares those totals out equally.

The procedure

Each interior angle

The interior angles of an n-gon add to (n - 2) \cdot 180^\circ. In a regular n-gon they are n equal angles, and n equal parts of a total are each the total divided by n:

\text{each interior angle} = \frac{(n - 2) \cdot 180^\circ}{n}

For a regular pentagon, n = 5. The angle sum is (5 - 2) \cdot 180^\circ = 540^\circ, so each angle is 540^\circ \div 5 = 108^\circ.

Notice what the argument used: that the angles are equal, and nothing about the sides. So the same formula gives each angle of any equiangular polygon, regular or not. A rectangle is an equiangular quadrilateral, and (4 - 2) \cdot 180^\circ \div 4 = 90^\circ.

Each exterior angle

At each vertex, the interior angle and the exterior angle form a linear pair, so

\text{exterior angle} = 180^\circ - \text{interior angle}

Equal interior angles therefore give equal exterior angles. The n exterior angles, one at each vertex, add to 360^\circ whatever n is, so each one is

\text{each exterior angle} = \frac{360^\circ}{n}

Figure 2At every vertex of a regular pentagon, the interior angle is 108^\circ and the exterior angle is 72^\circ.

For the regular pentagon, each exterior angle is 360^\circ \div 5 = 72^\circ, and 108^\circ + 72^\circ = 180^\circ, as Figure 2 shows.

This gives a second route to the interior angle, and it is often the quicker one: find the exterior angle first, then take it from 180^\circ.

\text{each interior angle} = 180^\circ - \frac{360^\circ}{n}

The two formulas always agree. By the distributive property, (n - 2) \cdot 180 = 180n - 360, and dividing each term by n gives

\frac{(n - 2) \cdot 180}{n} = \frac{180n - 360}{n} = 180 - \frac{360}{n}

In a regular n-gon:

n	angle sum	interior	exterior
3	180^\circ	60^\circ	120^\circ
4	360^\circ	90^\circ	90^\circ
5	540^\circ	108^\circ	72^\circ
6	720^\circ	120^\circ	60^\circ
8	1080^\circ	135^\circ	45^\circ
12	1800^\circ	150^\circ	30^\circ

Read down the table and the interior angle grows while the exterior angle shrinks. The second formula says why. As n grows, \displaystyle \frac{360}{n} gets smaller, so \displaystyle 180 - \frac{360}{n} gets larger, creeping up toward 180^\circ. It never reaches 180^\circ, because \displaystyle \frac{360}{n} is never 0. The smallest n is 3, which gives 60^\circ. So every interior angle of a regular polygon is at least 60^\circ and less than 180^\circ, and in particular every regular polygon is convex.

From an angle back to the number of sides

Run the exterior angle formula backwards. If each exterior angle measures e degrees, then \displaystyle e = \frac{360}{n}, and multiplying both sides by n and dividing by e gives

n = \frac{360}{e}

Given an interior angle instead, find the exterior angle first, as 180^\circ minus it. Solving \displaystyle \frac{(n - 2) \cdot 180}{n} = a directly also works, but needs more algebra to reach the same place.

The number of sides of a polygon is a whole number, and at least 3. So if \displaystyle \frac{360}{e} is not a whole number, there is no regular polygon with that angle. This is a real test, not a formality: 130^\circ is a perfectly good angle, and Example 4 shows that no regular polygon has it.

Worked examples

Example 1

A regular decagon, two ways

Find each interior angle and each exterior angle of a regular decagon.

Solution

A decagon has n = 10 sides. By the angle sum, the ten interior angles add to (10 - 2) \cdot 180^\circ = 1440^\circ. They are equal, so each one is 1440^\circ \div 10 = 144^\circ.

The ten exterior angles add to 360^\circ and are equal, so each is 360^\circ \div 10 = 36^\circ.

Check. 144^\circ + 36^\circ = 180^\circ, as it must, since each interior angle and its exterior angle form a linear pair. The check is also the second route: knowing the exterior angle is 36^\circ, the interior angle is 180^\circ - 36^\circ = 144^\circ with no angle sum needed.

Example 2

A regular heptagon, where the angle is not a whole number

Find each interior angle and each exterior angle of a regular heptagon, exactly.

Solution

A heptagon has n = 7 sides, and its angle sum is (7 - 2) \cdot 180^\circ = 900^\circ. Sharing that among seven equal angles gives \displaystyle \frac{900}{7} degrees for each. Since 900 = 7 \cdot 128 + 4, that is \displaystyle 128\frac{4}{7} degrees, about 128.57^\circ.

Each exterior angle is \displaystyle \frac{360}{7} degrees, and since 360 = 7 \cdot 51 + 3, that is \displaystyle 51\frac{3}{7} degrees, about 51.43^\circ.

The exact answers are the fractions. The decimals are rounded, so they add to 180.00 only because of the rounding; the fractions add to exactly 180: \displaystyle 128\frac{4}{7} + 51\frac{3}{7} = 179\frac{7}{7} = 180.

The angles are not whole numbers because 7 does not divide 360 evenly. A protractor would read about 129^\circ, which is close but not the angle.

Example 3

From an exterior angle to the number of sides

Each exterior angle of a regular polygon measures 40^\circ. How many sides does it have, and how big is each interior angle?

Solution

The exterior angles add to 360^\circ and are all 40^\circ, so there are

n = \frac{360}{40} = 9

of them, one at each vertex. The polygon has 9 sides: it is a regular nonagon. Each interior angle forms a linear pair with an exterior angle, so it is 180^\circ - 40^\circ = 140^\circ.

Check with the angle sum: (9 - 2) \cdot 180^\circ = 1260^\circ, and 9 \cdot 140 = 1260.

Example 4

From an interior angle, and an angle no regular polygon has

(a) Each interior angle of a regular polygon measures 165^\circ. How many sides does it have?

(b) Can each interior angle of a regular polygon measure 130^\circ?

Solution

(a) Go through the exterior angle. Each one is 180^\circ - 165^\circ = 15^\circ, and the exterior angles add to 360^\circ, so

n = \frac{360}{15} = 24

The polygon has 24 sides. Check: \displaystyle \frac{(24 - 2) \cdot 180}{24} = \frac{3960}{24} = 165.

(b) Each exterior angle would be 180^\circ - 130^\circ = 50^\circ, and then

n = \frac{360}{50} = 7.2

which is not a whole number. A polygon cannot have 7.2 sides, so no regular polygon has interior angles of 130^\circ. The table of angles shows the same thing from the other side: a regular heptagon’s angle is \displaystyle 128\frac{4}{7} degrees and a regular octagon’s is 135^\circ, and going from 7 sides to 8 jumps straight past 130^\circ.

Since the argument used only that the angles are equal, it shows more: no polygon at all has every angle equal to 130^\circ.

Example 5

When the interior angle is a multiple of the exterior angle

Each interior angle of a regular polygon is five times each exterior angle. How many sides does the polygon have, and what are its angles?

Solution

Call each exterior angle y degrees, so each interior angle is 5y degrees. An interior angle and its exterior angle form a linear pair, so they add to 180^\circ:

\begin{aligned} 5y + y &= 180 \\ 6y &= 180 \\ y &= 30 \end{aligned}

Each exterior angle is 30^\circ and each interior angle is 150^\circ. The number of sides is \displaystyle n = \frac{360}{30} = 12: a regular dodecagon, which agrees with the table.

Example 6

The diagonals from one vertex of a regular pentagon

In the regular pentagon ABCDE of Figure 3, the diagonals \overline{AC} and \overline{AD} are drawn. Find \angle BAC, \angle CAD and \angle DAE.

Figure 3The two diagonals from A cut the 108^\circ angle there into three angles of 36^\circ.

Solution

Each interior angle of a regular pentagon is 108^\circ.

Look at triangle ABC. Its sides \overline{AB} and \overline{BC} are sides of the pentagon, and a regular polygon’s sides are equal, so AB = BC. This is the first time the equal sides have mattered. The triangle is isosceles, so by the base angles theorem its base angles \angle BAC and \angle BCA are equal. The angle between the equal sides is \angle B = 108^\circ, so the two base angles share the remaining 180^\circ - 108^\circ = 72^\circ, and each is 36^\circ:

\angle BAC = 36^\circ

Triangle AED is the same: AE = ED and \angle E = 108^\circ, so \angle DAE = 36^\circ.

The three angles at A together make up the pentagon’s angle there, by angle addition, so

\angle CAD = 108^\circ - 36^\circ - 36^\circ = 36^\circ

All three are 36^\circ. The two diagonals from A cut the angle at A into three equal parts, and since every vertex of a regular pentagon looks the same, the same is true at every vertex.

Example 7

Two regular polygons, one with twice as many sides

One regular polygon has twice as many sides as another, and each of its interior angles is 30^\circ larger. What are the two polygons?

Solution

Work with exterior angles, since they have the simpler formula. Each interior angle is 180^\circ minus the exterior angle, so when the interior angle goes up by 30^\circ the exterior angle goes down by 30^\circ.

Say the first polygon has n sides. Its exterior angle is \displaystyle \frac{360}{n} degrees, and the second polygon’s is \displaystyle \frac{360}{2n} degrees. Writing the first over the same denominator, \displaystyle \frac{360}{n} = \frac{720}{2n}, the difference is

\frac{720}{2n} - \frac{360}{2n} = \frac{360}{2n} = \frac{180}{n}

So \displaystyle \frac{180}{n} = 30, and n = 6. The polygons are a regular hexagon and a regular dodecagon.

Check with the table: their interior angles are 120^\circ and 150^\circ, which differ by 30^\circ.

Boundaries

Regular needs both conditions. From four sides on, equal sides do not force equal angles, as the rhombus shows, and equal angles do not force equal sides, as the rectangle shows. Only for triangles is one condition enough.

The angle formula needs only the angles to be equal. Every equiangular n-gon has angles of \displaystyle \frac{(n - 2) \cdot 180^\circ}{n}, whether or not its sides match. So an angle measurement alone can never show that a polygon is regular; the sides have to be checked as well.

The number of sides must be a whole number, at least 3. Working backwards from an angle, a fractional answer for n is not an error in the arithmetic. It means no regular polygon has that angle.

The angles need not be whole numbers of degrees. When n does not divide 360 evenly, the exterior angle \displaystyle \frac{360}{n} is a fraction of a degree, and so is the interior angle. Keep the fraction; a rounded decimal is not the angle.

The sides must not cross. A five-pointed star drawn in one stroke has five equal sides and five equal point angles of 36^\circ, but its sides cross one another, so it is not a polygon in the sense of this lesson, and the formula, which gives 108^\circ for five sides, does not describe it. Stars like this are studied as a family of their own.

Flat surfaces only. Everything here comes from the angle sum, which comes from the triangle angle sum, which holds in the plane. On a sphere the totals, and so the shares, are different.

Where this goes next

Join the center of a regular n-gon to two neighboring vertices, and the angle at the center is \displaystyle \frac{360^\circ}{n}, the same number as the exterior angle. That is the subject of regular polygons and their circles, and a turn through that angle carries the polygon onto itself, which is its rotational symmetry. The same center gives the The apothem of a regular polygon, and with it the area of a regular polygon.

The interior angle decides how copies of a regular polygon can meet at a point. They fit around a point with no gap and no overlap only if their angles there add to exactly 360^\circ: six equilateral triangles, four squares or three regular hexagons do, while three regular pentagons leave a gap of 360^\circ - 3 \cdot 108^\circ = 36^\circ and four overlap. That is where Tessellations begin. Fold the copies up into space instead, and the angles at a corner must add to less than 360^\circ; together with the table above, that leaves room for only five regular solids.

The vertices of a regular polygon sit equally spaced around a circle. Placed in the complex plane with its center at 0 and one vertex at 1, they are exactly the nth roots of unity.

Exercises

Each angle of a regular polygon

Find each interior angle and each exterior angle.

A regular 16-gon.
A regular 20-gon.
A regular 11-gon. Give exact answers.
A regular 30-gon.

Working backwards

Each exterior angle of a regular polygon measures 20^\circ. How many sides does it have, and what is each interior angle?
Each interior angle of a regular polygon measures 156^\circ. How many sides does it have?
Each interior angle of a regular polygon measures 175^\circ. How many sides does it have?
Can each interior angle of a regular polygon measure 125^\circ? Explain.
Can each exterior angle of a regular polygon measure 70^\circ? Explain.

Angles in relation

Each interior angle of a regular polygon is three times each exterior angle. How many sides does it have?
Each interior angle of a regular polygon is 100^\circ more than each exterior angle. Find the number of sides and both angles.
Each interior angle of a regular polygon is said to be (4x + 12)^\circ and each exterior angle (x + 18)^\circ. Find x, then decide whether such a regular polygon exists.
One regular polygon has twice as many sides as another, and each of its interior angles is 18^\circ larger. Find both polygons.
The interior angles of a regular polygon add to 2160^\circ. How many sides does it have, and what is each angle, exactly?
Each interior angle of a regular polygon measures 170^\circ. What do all its interior angles add to?

Angles inside regular polygons

In a regular hexagon ABCDEF, the diagonal \overline{AC} is drawn. Find \angle BAC and \angle CAF.
In a regular octagon ABCDEFGH, find \angle BAC.
In a regular pentagon ABCDE, side \overline{AB} is extended past B and side \overline{DC} is extended past C, and the two extensions meet at P. Find \angle BPC.
For how many values of n is each interior angle of a regular n-gon a whole number of degrees?

Reasoning

Explain why a polygon with all its sides equal need not be regular, and give an example. Then explain why, for a triangle, equal sides are enough.
Show that \displaystyle \frac{(n - 2) \cdot 180}{n} and \displaystyle 180 - \frac{360}{n} are always equal, and explain why the second form is what you get by going through the exterior angle.
Explain why every interior angle of a regular polygon is at least 60^\circ and less than 180^\circ, and why it grows as the number of sides grows.
A hexagon has all six of its angles equal, and its sides, in order, measure 2, 4, 2, 4, 2 and 4. Find each angle, and explain why the hexagon is not regular.

Answers

Interior 157.5^\circ, exterior 22.5^\circ. The exterior angle is 360 \div 16 = 22.5, and the interior angle is 180 - 22.5 = 157.5.
Interior 162^\circ, exterior 18^\circ. 360 \div 20 = 18, and 180 - 18 = 162.
Interior \displaystyle \frac{1620}{11} degrees, which is \displaystyle 147\frac{3}{11} degrees (about 147.27^\circ); exterior \displaystyle \frac{360}{11} degrees, which is \displaystyle 32\frac{8}{11} degrees (about 32.73^\circ). The angle sum is 9 \cdot 180 = 1620, shared among 11 angles.
Interior 168^\circ, exterior 12^\circ. 360 \div 30 = 12, and 180 - 12 = 168.
18 sides, each interior angle 160^\circ. 360 \div 20 = 18, and 180 - 20 = 160.
15 sides. Each exterior angle is 180 - 156 = 24 degrees, and 360 \div 24 = 15.
72 sides. Each exterior angle is 180 - 175 = 5 degrees, and 360 \div 5 = 72.
No. Each exterior angle would be 55^\circ, and \displaystyle 360 \div 55 = \frac{72}{11}, which is not a whole number.
No. \displaystyle 360 \div 70 = \frac{36}{7}, which is not a whole number. The angle falls between a regular pentagon’s 72^\circ and a regular hexagon’s 60^\circ.
8 sides. With exterior angle y, the linear pair gives 3y + y = 180, so y = 45, and 360 \div 45 = 8.
9 sides; interior 140^\circ, exterior 40^\circ. With exterior angle y, the interior is 180 - y, and (180 - y) - y = 100 gives y = 40. Then 360 \div 40 = 9.
x = 30, and no such regular polygon exists. The two angles form a linear pair, so 5x + 30 = 180. They would be 132^\circ and 48^\circ, and 360 \div 48 = 7.5 is not a whole number.
A regular decagon and a regular 20-gon, with interior angles 144^\circ and 162^\circ. The exterior angles \displaystyle \frac{360}{n} and \displaystyle \frac{360}{2n} differ by \displaystyle \frac{180}{n}, so \displaystyle \frac{180}{n} = 18 and n = 10.
14 sides, each angle \displaystyle \frac{1080}{7} degrees, which is \displaystyle 154\frac{2}{7} degrees. (n - 2) \cdot 180 = 2160 gives n - 2 = 12, and \displaystyle 2160 \div 14 = \frac{1080}{7}.
6120^\circ. Each exterior angle is 10^\circ, so n = 360 \div 10 = 36, and the angle sum is (36 - 2) \cdot 180 = 6120.
\angle BAC = 30^\circ and \angle CAF = 90^\circ. Triangle ABC has AB = BC and \angle B = 120^\circ, so each base angle is (180 - 120) \div 2 = 30. The angle at A is 120^\circ, and 120 - 30 = 90.
22.5^\circ. Each angle of a regular octagon is 135^\circ, and triangle ABC is isosceles with AB = BC, so \angle BAC = (180 - 135) \div 2 = 22.5.
36^\circ. \angle PBC and \angle PCB are exterior angles of the pentagon, 72^\circ each, so in triangle BPC the third angle is 180 - 72 - 72 = 36.
22. Each interior angle is \displaystyle 180 - \frac{360}{n} degrees, a whole number exactly when \displaystyle \frac{360}{n} is, that is, when n divides 360. Since 360 = 2^3 \cdot 3^2 \cdot 5, it has 4 \cdot 3 \cdot 2 = 24 divisors, and leaving out 1 and 2 (a polygon has at least 3 sides) leaves 22: 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360.
Equal sides say nothing about the angles once there are four or more sides. A rhombus with angles of 60^\circ and 120^\circ has four equal sides but unequal angles, so it is not regular. In a triangle, each pair of equal sides makes the angles opposite them equal, by the base angles theorem, so three equal sides make all three angles equal, and the triangle is regular.
By the distributive property (n - 2) \cdot 180 = 180n - 360, and dividing each term by n gives \displaystyle 180 - \frac{360}{n}. Going through the exterior angle: the n equal exterior angles add to 360^\circ, so each is \displaystyle \frac{360}{n} degrees, and the interior angle forms a linear pair with it, so it is \displaystyle 180 - \frac{360}{n} degrees.
Each interior angle is \displaystyle 180 - \frac{360}{n} degrees. Since \displaystyle \frac{360}{n} is positive, the angle is less than 180^\circ. As n increases, \displaystyle \frac{360}{n} decreases, so the angle increases. The smallest possible n is 3, giving 180 - 120 = 60, so no regular polygon has a smaller angle.
Each angle is 120^\circ, and the hexagon is not regular. Its six equal angles add to (6 - 2) \cdot 180 = 720 degrees, so each is 720 \div 6 = 120. Its sides are not all equal, so it is equiangular but not equilateral. Such a hexagon exists: cut an equilateral triangle of side 2 off each corner of an equilateral triangle of side 8.