Decagon - Definition, Sides, Angles, Diagonals, and Area Formula

The Ten-sided Coin You May Have Held Without Counting Its Edges

Belize once minted a fifty-cent coin with exactly ten straight edges, so a person could tell it apart in a pocket by feel alone. That coin is a decagon, and its ten sides do a job a circle cannot.

A decagon is a polygon with ten straight sides and ten vertices; the prefix "deca" means ten. A regular decagon has all ten sides equal and all ten angles equal, while an irregular decagon has sides or angles of differing measure. Like every polygon, its interior angles obey the sum rule, which for ten sides gives $1440°$. The decagon sits alongside the hexagon and octagon as a named member of the polygons family.

By the end you will know why a regular decagon's angles are $144°$, how many diagonals it has, and where its area formula comes from. The centre-to-side distance marked above is the apothem, and it is the key to the decagon's area.

Angles Of A Decagon

Every decagon, regular or not, has interior angles that sum to the same total. Use the polygon angle-sum formula with $n = 10$:

$$\text{Sum of interior angles} = (n-2)\times 180° = (10-2)\times 180° = 8 \times 180° = 1440°$$

Here $n = 10$ is the side count, and $(n-2) = 8$ is the number of triangles that diagonals from one vertex carve the decagon into. For a regular decagon, the ten equal angles share that total:

$$\text{Each interior angle} = \frac{1440°}{10} = 144°$$

The exterior angle at each vertex is the supplement, $180° - 144° = 36°$, and the ten exterior angles sum to $360°$, as they do for every polygon. These are the same interior and exterior angles defined in interior angles.

Why does the interior angle stop growing as sides increase? As a polygon gains sides it gets rounder, so each corner opens wider toward the straight $180°$ line without ever reaching it. A decagon's $144°$ is closer to that limit than a hexagon's $120°$, which is exactly what "more sides looks more like a circle" means numerically.

Properties Of A Regular Decagon

A regular decagon's symmetry gives it a clean set of properties worth knowing before any calculation.

• 10 sides, 10 vertices, 10 lines of symmetry. It maps onto itself under rotations of $36°$.

• Interior angle $144°$, exterior angle $36°$. The interior-angle sum is $1440°$.

• 35 diagonals. Using $\dfrac{n(n-3)}{2}$ with $n = 10$: $\dfrac{10 \times 7}{2} = 35$.

• It is convex. Every interior angle ($144°$) is below $180°$, so a regular decagon is a convex polygon.

• It splits into 10 equal isosceles triangles from the centre, which powers the area formula below.

Deriving The Area Of A Regular Decagon

Rather than memorise the area formula, build it from the apothem, the perpendicular distance from the centre to the middle of a side.

Slice the regular decagon from its centre to every vertex. This produces 10 identical isosceles triangles, each with base $s$ (a side) and height $a$ (the apothem). One triangle has area $\tfrac{1}{2},s,a$, so the whole decagon is:

$$A = 10 \times \tfrac{1}{2},s,a = \tfrac{1}{2},(10s),a = \tfrac{1}{2},P,a$$

where $P = 10s$ is the perimeter. This $A = \tfrac{1}{2},P,a$ is the universal area rule for any regular polygon. For the regular decagon, writing the apothem in terms of the side and simplifying gives the side-only formula:

$$A = \tfrac{5}{2},a^2,\sqrt{5 + 2\sqrt{5}} \approx 7.694,a^2$$

Here $a$ is the side length, the factor $\tfrac{5}{2}\sqrt{5+2\sqrt5}$ is a fixed constant for every regular decagon, and the result is in square units of whatever unit the side uses.

Examples Of Decagon

Example 1

Find the sum of the interior angles of a decagon and each angle of a regular decagon.

Sum of interior angles:

$$(10-2)\times 180° = 8 \times 180° = 1440°$$

Each angle of a regular decagon:

$$\frac{1440°}{10} = 144°$$

The interior angles total $1440°$, and each regular-decagon angle is $144°$.

Example 2

A student finds each interior angle of a regular decagon as $\dfrac{1440°}{8} = 180°$. Spot the error.

A natural first move is to divide the angle sum by the number of triangles, $8$. Try it: $1440° \div 8 = 180°$. But a $180°$ "angle" is a straight line, which a real corner cannot be, so that absurd result flags a wrong divisor.

Divide by the number of angles instead, which is $10$:

$$\frac{1440°}{10} = 144°$$

Each interior angle is $144°$. The $(n-2)$ counts triangles; the divisor for each angle is always $n$.

Example 3

A regular decagon has a side length of $6$ cm. Find its perimeter and area.

Perimeter is ten equal sides:

$$P = 10 \times 6 = 60 \text{ cm}$$

Area uses the side-only formula:

$$A = \tfrac{5}{2},a^2\sqrt{5+2\sqrt5} = \tfrac{5}{2}(6)^2(3.0777) \approx \tfrac{5}{2}(36)(3.0777) \approx 277.0 \text{ cm}^2$$

The perimeter is $60$ cm and the area is about $277.0$ cm².

Example 4

How many diagonals does a decagon have?

Use the diagonal formula for an $n$-gon:

$$\frac{n(n-3)}{2} = \frac{10(10-3)}{2} = \frac{10 \times 7}{2} = \frac{70}{2} = 35$$

A decagon has 35 diagonals.

Example 5

A regular decagon has an apothem of $9$ cm and a side of $6$ cm. Find its area using the perimeter-apothem formula.

Perimeter:

$$P = 10 \times 6 = 60 \text{ cm}$$

Area:

$$A = \tfrac{1}{2},P,a = \tfrac{1}{2}\times 60 \times 9 = 270 \text{ cm}^2$$

The area is $270$ cm². It sits close to the side-only result, the small gap coming from rounding the apothem.

Example 6

A designer wants a regular decagon window and asks what angle to cut where two frame pieces meet at a corner. Find that angle.

The angle inside each corner of a regular decagon is its interior angle.

$$\text{Interior angle} = \frac{(10-2)\times 180°}{10} = \frac{1440°}{10} = 144°$$

Each mitre corner must open to $144°$, so each of the two frame pieces is cut at half that turn from straight, $\tfrac{180° - 144°}{2} = 18°$. This is the same reasoning a carpenter uses for any regular polygon frame.

Where The Decagon Earns Its Keep: Shapes Read By Feel And By Eye

The decagon shows up wherever ten equal sides give a shape that is round enough to be pleasing yet still made of straight, cuttable edges.

• Coins. Some nations mint ten-sided coins so people can identify a denomination by touch, without looking, the same reason the UK's old threepenny bit and other coins used non-circular outlines.

• Design and architecture. Decagonal floor tiles, decorative wall clocks, gazebos, and stained-glass panels use the ten-sided outline because it approaches a circle while staying buildable from straight segments.

• Why ten sides. The destination is near-round with straight edges. A decagon is close enough to a circle to look refined, yet every side is a straight cut a machine or a mason can make, which a true circle never allows.

The idea that non-circular coin edges aid recognition is well documented in the design of circulating coinage, where shape and edge are chosen so the coin is told apart by touch.

Mistakes To Watch For In A Decagon

Mistake 1: Dividing the angle sum by $8$ instead of $10$

Where it slips in: Finding each interior angle of a regular decagon right after computing the $1440°$ sum.

Don't do this: Write $\dfrac{1440°}{8} = 180°$, confusing the triangle count $(n-2)$ with the angle count $n$.

The correct way: The sum $1440°$ is shared among the ten angles, so divide by $10$ to get $144°$. The student who blurs the two numbers gets a meaningless $180°$ corner and should read that impossible result as a signal to recheck the divisor.

Mistake 2: Using the regular-decagon area formula on an irregular decagon

Where it slips in: Applying $A = \tfrac{5}{2}a^2\sqrt{5+2\sqrt5}$ to a ten-sided shape whose sides are not all equal.

Don't do this: Plug one side length into the regular formula when the ten sides differ.

The correct way: The side-only formula and the apothem exist only for regular decagons. For an irregular one, split it into triangles and add their areas. The memoriser who stored "decagon area $= \tfrac{5}{2}a^2\sqrt{5+2\sqrt5}$" as a fact applies it to the first lopsided ten-sided plot and reports the wrong area.

Mistake 3: Miscounting the diagonals

Where it slips in: Trying to count all diagonals of a decagon by drawing them by hand.

Don't do this: Draw and tally, then lose track past twenty and guess.

The correct way: Use $\dfrac{n(n-3)}{2}$. Each of the $10$ vertices connects to $n-3 = 7$ non-adjacent vertices, and dividing by $2$ removes the double count, giving $35$.

Key Takeaways

• A decagon has 10 sides; its interior angles sum to $1440°$.

• A regular decagon has each interior angle $144°$, each exterior angle $36°$, and 35 diagonals.

• The area is $A = \tfrac{5}{2}a^2\sqrt{5+2\sqrt5}$, derived from $A = \tfrac{1}{2},P,a$ by splitting the decagon into 10 triangles.

• The regular formula applies only when all ten sides are equal; irregular decagons are split into triangles instead.

• A regular decagon is convex, with ten lines of symmetry.

A practical next step

Practice these problems to solidify your understanding. Sketch each decagon and label its angle, side, and apothem before calculating.

1. A regular decagon has a side of $4$ cm. Find its area. (Answer to Question 1: $\tfrac{5}{2}(4)^2\sqrt{5+2\sqrt5} \approx 123.1$ cm².)

2. Find each exterior angle of a regular decagon. (Answer to Question 2: $360° \div 10 = 36°$.)

To work through decagons and other polygons with a teacher, explore Bhanzu's geometry tutor, our middle school math tutor sessions, or math classes online. To see a trainer derive the decagon area live, you can book a free demo class.

Read More

• Types of polygon — where the regular decagon sits among all polygon classes.

• Sum of angles in a polygon — the $(n-2)\times 180°$ rule the $1440°$ total comes from.

• Pentagon shape — the five-sided polygon with $108°$ interior angles.

• Irregular polygons — how decagons behave when their sides are not all equal.

• Exterior angles of a polygon — the $36°$ decagon exterior angle inside the $360°$ rule.