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Pentagon

Pentagon

A polygon with five sides.

 

An irregular pentagon with five unequal sides and angles, wider at the top and narrowing toward the bottom right.

Pentagon

A regular pentagon (5-sided polygon) with equal sides and angles, shown as a plain white geometric figure.

Regular Pentagon

Key Formula

S=(n−2)×180°where n=5,S=540°S = (n - 2) \times 180°\quad\text{where } n = 5,\quad S = 540°
Where:
  • SS = Sum of the interior angles of the pentagon
  • nn = Number of sides (5 for a pentagon)

Worked Example

Problem: A regular pentagon has a side length of 10 cm. Find the measure of each interior angle and the perimeter.
Step 1: Find the sum of all interior angles using the polygon angle-sum formula with n = 5.
S=(5−2)×180°=3×180°=540°S = (5 - 2) \times 180° = 3 \times 180° = 540°
Step 2: Since a regular pentagon has all angles equal, divide the total by 5 to get each interior angle.
Each angle=540°5=108°\text{Each angle} = \frac{540°}{5} = 108°
Step 3: Find the perimeter by multiplying the side length by the number of sides.
P=5×10 cm=50 cmP = 5 \times 10\text{ cm} = 50\text{ cm}
Answer: Each interior angle of the regular pentagon measures 108°, and its perimeter is 50 cm.

Another Example

Problem: Four angles of a pentagon measure 100°, 110°, 120°, and 95°. Find the fifth angle.
Step 1: Recall that the interior angles of any pentagon sum to 540°.
S=540°S = 540°
Step 2: Add the four known angles.
100°+110°+120°+95°=425°100° + 110° + 120° + 95° = 425°
Step 3: Subtract from 540° to find the missing angle.
540°−425°=115°540° - 425° = 115°
Answer: The fifth angle measures 115°.

Frequently Asked Questions

How many diagonals does a pentagon have?
A pentagon has 5 diagonals. You can calculate this with the diagonal formula: n(n − 3)/2 = 5(5 − 3)/2 = 5. A diagonal connects two non-adjacent vertices, so from each of the 5 vertices you can draw 2 diagonals, but you divide by 2 to avoid counting each one twice.
What does a regular pentagon look like and where do you see pentagons in real life?
A regular pentagon has five equal sides and five equal angles of 108° each, giving it a balanced, slightly rounded appearance. The most famous real-world example is the Pentagon building in Washington, D.C. You also see pentagons on soccer balls (the black patches) and in cross-sections of certain fruits like okra.

Pentagon vs. Hexagon

A pentagon has 5 sides and interior angles summing to 540°, while a hexagon has 6 sides and interior angles summing to 720°. Each interior angle of a regular pentagon is 108°; each interior angle of a regular hexagon is 120°. Adding one side increases the angle sum by 180°.

Why It Matters

The pentagon appears frequently in geometry, architecture, and nature. Understanding its angle-sum property (540°) lets you solve for unknown angles in five-sided figures. Regular pentagons also connect to the golden ratio — the ratio of a diagonal to a side in a regular pentagon equals (1 + √5)/2 — linking this simple shape to deeper mathematical ideas.

Common Mistakes

Mistake: Using 180° × 5 = 900° for the interior angle sum instead of the correct formula (n − 2) × 180°.
Correction: Always subtract 2 from the number of sides first. For a pentagon: (5 − 2) × 180° = 540°, not 900°.
Mistake: Assuming every pentagon has equal sides and equal angles.
Correction: Only a regular pentagon has all sides and angles equal. Irregular pentagons can have sides and angles of different sizes, though their angles still sum to 540°.

Related Terms

  • Polygon — General term for any closed multi-sided shape
  • Side of a Polygon — Each straight segment forming the pentagon
  • Regular Polygon — Polygon with all sides and angles equal
  • Hexagon — Six-sided polygon, one more side than a pentagon
  • Quadrilateral — Four-sided polygon, one fewer side than a pentagon
  • Interior Angle — Angle formed inside the pentagon at each vertex
  • Diagonal — Line segment connecting non-adjacent vertices