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Decagon

Decagon

A polygon with ten sides.

 

An irregular decagon (10-sided polygon) with an uneven, arrow-like outline showing all ten connected sides.

Decagon

A regular decagon (10-sided polygon) with equal sides and equal interior angles.

Regular Decagon

Key Formula

S=(n−2)×180∘where n=10,S=1440∘S = (n - 2) \times 180^\circ \quad \text{where } n = 10, \quad S = 1440^\circ
Where:
  • SS = Sum of the interior angles of the decagon
  • nn = Number of sides (10 for a decagon)

Worked Example

Problem: A regular decagon has a side length of 8 cm. Find the measure of each interior angle and calculate the perimeter.
Step 1: Find the sum of all interior angles using the polygon angle-sum formula.
S=(10−2)×180∘=8×180∘=1440∘S = (10 - 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ
Step 2: Since the decagon is regular, all ten interior angles are equal. Divide the total by 10.
Each angle=1440∘10=144∘\text{Each angle} = \frac{1440^\circ}{10} = 144^\circ
Step 3: Find the perimeter by multiplying the side length by the number of sides.
P=10×8=80 cmP = 10 \times 8 = 80 \text{ cm}
Answer: Each interior angle of the regular decagon measures 144°, and the perimeter is 80 cm.

Another Example

Problem: Find the measure of each exterior angle of a regular decagon.
Step 1: The exterior angles of any convex polygon always sum to 360°.
Sum of exterior angles=360∘\text{Sum of exterior angles} = 360^\circ
Step 2: For a regular decagon, all exterior angles are equal. Divide 360° by 10.
Each exterior angle=360∘10=36∘\text{Each exterior angle} = \frac{360^\circ}{10} = 36^\circ
Step 3: Verify: each interior angle plus its exterior angle should equal 180°.
144∘+36∘=180∘✓144^\circ + 36^\circ = 180^\circ \checkmark
Answer: Each exterior angle of a regular decagon measures 36°.

Frequently Asked Questions

How many diagonals does a decagon have?
A decagon has 35 diagonals. You can find this using the formula d=n(n−3)2d = \frac{n(n-3)}{2}, which gives 10(10−3)2=702=35\frac{10(10-3)}{2} = \frac{70}{2} = 35.
What does a regular decagon look like?
A regular decagon looks like a slightly rounded ten-sided shape, almost approaching a circle. All ten sides are the same length, and every interior angle is exactly 144°. You can see this shape on some coins, like the Australian 50-cent coin.

Decagon (10 sides) vs. Dodecagon (12 sides)

Why It Matters

Decagons appear in architecture, tiling patterns, and coin designs around the world. Understanding their angle properties helps you work with any polygon, since the same angle-sum formula applies to shapes with any number of sides. The regular decagon also has a deep connection to the golden ratio — the ratio of its diagonal to its side equals the golden ratio ϕ≈1.618\phi \approx 1.618.

Common Mistakes

Mistake: Confusing a decagon (10 sides) with a dodecagon (12 sides) or a nonagon (9 sides).
Correction: Remember that the prefix "deca-" means ten (as in "decade" = ten years). A dodecagon has 12 sides, and a nonagon has 9.
Mistake: Assuming that every decagon has equal sides and angles.
Correction: Only a regular decagon has all sides and all angles equal. An irregular decagon can have sides and angles of different sizes, as long as it still has exactly ten sides.

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