A pentagon is the simplest polygon after the triangle and quadrilateral whose interior angles add to more than two right angles. With five sides, the sum reaches $540°$ — three times that of a triangle.
The Formal Definition
A pentagon is a polygon with five sides, five vertices, and five interior angles. The word comes from the Greek pente ("five") and gonia ("angle").
The interior angles of any pentagon always sum to:
$$(n - 2) \times 180° = (5 - 2) \times 180° = 540°.$$
This holds whether the pentagon is regular or irregular, convex or concave — the sum is fixed by the number of sides.
A pentagon is convex if every interior angle is less than $180°$ (and every diagonal lies inside the shape). A pentagon is concave if at least one interior angle is greater than $180°$ — a "reflex" angle that bends the shape inward.
Quick reference.
Definition: five-sided closed polygon.
Sides: 5. Vertices: 5. Diagonals: 5.
Sum of interior angles: $540°$.
Each interior angle (regular): $108°$.
Each exterior angle (regular): $72°$.
Area (regular, side $a$): $\dfrac{1}{4}\sqrt{5(5 + 2\sqrt{5})}, a^{2} \approx 1.720, a^{2}$.
Perimeter (regular): $5a$.
Grade introduced: CCSS-M 5.G.B (polygon classification); NCERT Class 8 — Understanding Quadrilaterals (incl. pentagon angle sum).
Types of Pentagon
Pentagons fall into a small family of named subtypes.
Regular pentagon. All five sides equal, all five angles equal ($108°$ each). Five lines of symmetry. The ratio of its diagonal to its side is the golden ratio $\varphi = (1 + \sqrt{5})/2 \approx 1.618$.
Irregular pentagon. Sides and/or angles not all equal. The most common kind in real-world shapes.
Convex pentagon. Every interior angle less than $180°$. All diagonals lie inside.
Concave pentagon. At least one interior angle greater than $180°$. At least one diagonal lies outside the shape.
Cyclic pentagon. All five vertices lie on a single circle. Every regular pentagon is cyclic; most irregular pentagons are not.
These categories overlap — a regular pentagon is regular and convex and cyclic at the same time.
Properties of a Regular Pentagon
The regular pentagon is the most-asked-about because it has the cleanest properties.
Property | Value |
|---|---|
Number of sides | $5$ |
Number of vertices | $5$ |
Number of diagonals | $5$ — confirmed by $\dfrac{n(n-3)}{2} = \dfrac{5 \times 2}{2}$ |
Interior angle (each) | $108°$ |
Exterior angle (each) | $72°$ |
Interior angle sum | $540°$ |
Lines of symmetry | $5$ |
Rotational symmetry order | $5$ |
Diagonal-to-side ratio | $\varphi = \dfrac{1 + \sqrt{5}}{2}$ |
A property worth noting: the five diagonals of a regular pentagon form a smaller regular pentagon (and a five-pointed star) inside it. The pattern repeats indefinitely — a regular pentagon is one of the cleanest naturally-occurring fractals.
Area, Perimeter, and the Golden Ratio
Perimeter of a regular pentagon (side $a$):
$$P = 5a.$$
Area of a regular pentagon (side $a$) — derived from dividing the pentagon into five congruent isosceles triangles from the centre:
$$A = \frac{1}{4}\sqrt{5(5 + 2\sqrt{5})}, a^{2} \approx 1.720, a^{2}.$$
A friendlier form using the apothem $h$ (the distance from the centre to the midpoint of a side):
$$A = \frac{1}{2} \times P \times h = \frac{5ah}{2}.$$
The apothem of a regular pentagon with side $a$ is $h = \dfrac{a}{2 \tan(36°)} \approx 0.688, a$.
Three Worked Examples — Quick, Standard, Stretch
Quick. What is the sum of the interior angles of a pentagon?
Use the polygon angle-sum formula $(n - 2) \times 180°$ with $n = 5$:
$$(5 - 2) \times 180° = 540°.$$
Final answer: $540°$.
Standard (Wrong Path First — The Tempting Shortcut). Four interior angles of a pentagon are $100°, 110°, 120°, 130°$. Find the fifth angle.
The wrong path. A student divides $540°$ by $5$ to get $108°$ and writes that as the answer.
The flaw: $108°$ is the value of each interior angle of a regular pentagon. This pentagon is irregular — its angles are clearly not all equal. The fifth angle must be found by subtraction, not by dividing the total.
The rescue. The five interior angles sum to $540°$. Subtract the known four from $540°$:
$$540° - (100° + 110° + 120° + 130°) = 540° - 460° = 80°.$$
Final answer: $80°$.
The lesson — the angle-sum rule is fixed at $540°$, but the individual angles only divide evenly when the pentagon is regular. When the problem gives unequal angles, subtract; don't average.
Stretch. Find the area of a regular pentagon with side $6$ cm. Round to two decimal places.
Use $A \approx 1.720, a^{2}$ with $a = 6$:
$$A \approx 1.720 \times 36 = 61.92 \text{ cm}^{2}.$$
For a tighter answer, use the exact form $A = \tfrac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \times 36$. Compute $5 + 2\sqrt{5} = 5 + 2(2.2361) \approx 9.4721$. Then $\sqrt{5 \times 9.4721} = \sqrt{47.36} \approx 6.882$. So $A \approx \tfrac{1}{4} \times 6.882 \times 36 \approx 61.94$ cm².
Final answer: $A \approx 61.94$ cm².
This is the version of pentagon problem that shows up in NCERT Class 8 Mensuration and in CBSE board questions on regular polygon area.
Where Pentagons Appear — Beyond the Shape Chart
A few places this shape carries weight:
Architecture. The Pentagon building in Arlington, Virginia is a regular pentagon — chosen partly because the original 1941 site was already a pentagonal junction of roads.
Football. A soccer ball is the classical truncated icosahedron — $20$ hexagons and $12$ regular pentagons. The pentagons are what stop the ball from collapsing flat.
Crystal structures. Many quasicrystals (a 2011 Nobel discovery) show five-fold symmetry — impossible in a regular crystal lattice, but real in nature.
Plant biology. Apples, pears, and many flowers (okra, hibiscus) show five-fold symmetry in their cross-section.
Pentagram in art. The regular five-pointed star drawn inside a pentagon is the basis of the pentagram, found in religious art across cultures and famously in the Vitruvian Man drawing by Leonardo da Vinci.
The construction of a regular pentagon with only compass and straightedge appears as Proposition $11$ in Book IV of Euclid's Elements — among the most elegant constructions in classical geometry, leaning entirely on the golden ratio.
Tripping Points to Avoid in Pentagon
Mistake 1: Dividing the angle sum by $5$ for an irregular pentagon
Where it slips in: Student sees "find the missing angle" and divides $540°$ by $5$.
Don't do this: Use the regular-pentagon shortcut on an irregular pentagon.
The correct way: Subtract the sum of the known angles from $540°$. The angle-sum formula gives the total, not the individual values, unless every angle is known to be equal.
Mistake 2: Forgetting that a polygon must be closed
Where it slips in: Drawing five line segments that meet but don't close back to the starting point and calling it a pentagon.
Don't do this: Skip the closure step.
The correct way: A pentagon is a closed polygon — start and end at the same vertex. An open zig-zag of five segments is not a pentagon.
Mistake 3: Counting a five-pointed star as a pentagon
Where it slips in: Confusing a pentagram (five-pointed star) with a pentagon (five-sided figure).
Don't do this: Equate the two.
The correct way: A pentagon has five straight sides forming a closed curve with no self-intersection. A pentagram is a self-intersecting figure with $10$ visible edges and $5$ interior crossings. They're related (the pentagram lives inside the regular pentagon) but distinct shapes.
A real-world version of the mistake. In the construction of The Pentagon (Washington, completed 1943), the original architectural drawings showed a pentagon with all five sides equal — a regular pentagon. But the site forced one side to be longer to fit the road geometry. Engineers had to redo the structural-load calculations because every interior beam length depended on the equal-side assumption. Treating a real-world irregular pentagon as regular is not a textbook slip; it's an engineering risk.
Conclusion
A pentagon is a five-sided closed polygon — five sides, five vertices, five interior angles summing to $540°$.
Regular pentagons have all sides and angles equal ($108°$ each); irregular pentagons do not.
The diagonal-to-side ratio of a regular pentagon equals the golden ratio $\varphi$.
Area of a regular pentagon with side $a$ is $\approx 1.720, a^{2}$.
The most common mistake is averaging $540°/5$ for an irregular pentagon — subtract, don't divide, when the angles are unequal.
Five Minutes of Practice
The interior angle sum of an octagon is $1080°$. Verify using $(n-2) \times 180°$.
Three angles of a pentagon are $90°, 100°, 110°$. The other two are equal — find their measure.
Find the perimeter of a regular pentagon with side $7$ cm.
If problem 2 gave you $108°$, return to Mistake 1 above.
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