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

TL;DR

A decagon is a ten-sided polygon whose interior angles sum to 1440°; in a regular decagon each interior angle is 144° and each exterior angle is 36°. This article defines the decagon, derives its area formula, counts its 35 diagonals, and works through examples.

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

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°. 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.

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:

Sum of interior angles=(n−2)×180°=(10−2)×180°=8×180°=1440°

For a regular decagon, the ten equal angles share that total:

Each interior angle=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.

Properties Of A Regular Decagon

A regular decagon's symmetry gives it a clean set of properties:

• 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 n(n−3)/2 with n=10: 10×7/2=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 center, which powers the area formula.

Deriving The Area Of A Regular Decagon

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

Slice the regular decagon from its centre to every vertex to produce 10 identical isosceles triangles. The area is:

A=10×(1/2)s*a = (1/2)(10s)a = (1/2)Pa

where P=10s is the perimeter. This A=(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=(5/2)a²√(5 + 2√5) ≈ 7.694a²

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)×180°=8×180°=1440°

Each angle of a regular decagon:

1440°/10=144°

Example 2

A student finds each interior angle of a regular decagon as 1440°/8 = 180°. Spot the error.

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

1440°/10=144°

Example 3

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

Perimeter:

P=10×6=60 cm

Area:

A=(5/2)(6)²√(5 + 2√5) ≈ 277.0 cm²

Example 4

How many diagonals does a decagon have?

Use the diagonal formula for an n-gon:

d(n)=n(n−3)/2 = 10(10−3)/2 = 35

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.

Area:

A=(1/2)P*a = (1/2)×60×9 = 270 cm²

Example 6

A designer wants a regular decagon window and asks what angle to cut.

Interior angle:

= (10−2)×180°/10 = 144°

Each mitre corner must open to 144°, cut at half that turn: 180°−144°/2 = 18°

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=(5/2)a²√(5 + 2√5), derived from A=(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.