What Is a Pentagon?
A pentagon is a polygon with 5 sides and 5 vertices. The name comes from Greek penta (five) + gonia (angle).
Sum of interior angles of any pentagon:
$$(n - 2) \times 180° = (5 - 2) \times 180° = 540°$$
This is true for any pentagon — convex or concave, regular or irregular.
Types of Pentagons
Regular Pentagon
All 5 sides equal length; all 5 angles equal (each 108°). The most symmetric pentagon. Five axes of symmetry; rotational symmetry of order 5.
Irregular Pentagon
Sides and/or angles unequal. Most pentagons in real-world geometry problems are irregular.
Convex Pentagon
All interior angles < 180°. All diagonals lie inside the pentagon. Most common type.
Concave Pentagon
At least one interior angle > 180° (a "dent"). At least one diagonal exits the pentagon.
Properties of a Regular Pentagon
All sides equal: each of length $s$.
All interior angles equal: each $108°$.
All exterior angles equal: each $72°$.
Sum of interior angles: $540°$.
5 axes of symmetry — each through one vertex and the midpoint of the opposite side.
5-fold rotational symmetry — rotates onto itself every $72°$.
Diagonals form a pentagram (a 5-pointed star) — when all 5 diagonals are drawn.
Diagonal-to-side ratio is the golden ratio $\varphi = \dfrac{1 + \sqrt{5}}{2} \approx 1.618$.
Area Formulas
Regular Pentagon — Exact Formula
$$A = \frac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \cdot s^2$$
The constant $\tfrac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \approx 1.72048$.
Practical form (rounded):
$$A \approx 1.72 s^2$$
Regular Pentagon — Using Apothem
The apothem $a$ is the distance from the centre to the midpoint of a side.
$$A = \frac{1}{2} \cdot P \cdot a = \frac{5sa}{2}$$
Useful when the apothem is known directly.
Irregular Pentagon
No single formula. Divide into triangles (one common approach: connect each vertex to one chosen vertex, producing 3 triangles), compute each triangle's area, sum them.
Perimeter of a Pentagon
For any pentagon:
$$P = s_1 + s_2 + s_3 + s_4 + s_5$$
For a regular pentagon with side $s$:
$$P = 5s$$
Three Worked Examples — Quick, Standard, Stretch
Quick — Perimeter
A regular pentagon has side $7$ cm. Find its perimeter.
$P = 5 \times 7 = 35$ cm.
Standard — Area Using Side
A regular pentagon has side $6$ m. Find its area.
$$A \approx 1.72 \times 6^2 = 1.72 \times 36 \approx 61.94 \text{ m}^2$$
Exactly: $A = \tfrac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \cdot 36$.
Stretch — Area Using Apothem
A regular pentagon has side $4$ cm and apothem $2.75$ cm. Find its area.
$$A = \frac{5sa}{2} = \frac{5 \cdot 4 \cdot 2.75}{2} = 27.5 \text{ cm}^2$$
Why Does the Pentagon Matter? (The Real-World GROUND)
"The pentagon contains the golden ratio." — geometric truth.
Pentagons appear in nature, architecture, and science:
The Pentagon (US Department of Defense headquarters) is a regular-pentagon-shaped building, completed 1943.
Sea stars (starfish) have 5-fold radial symmetry — pentagonal.
Flowers — many flower species have 5 petals arranged in a pentagonal pattern (roses, hibiscus, lilies, columbines).
Pentaprisms — used in cameras and surveying instruments to deflect light by exactly 90° regardless of angle of incidence.
Crystalline structures — pentagonal symmetry was once thought impossible in crystals; Dan Shechtman's 1982 discovery of quasicrystals with 5-fold symmetry overturned that and won him the 2011 Nobel Prize in Chemistry.
Soccer balls — the classic 32-panel truncated icosahedron has 12 pentagonal panels (and 20 hexagonal).
Phi (the golden ratio) appears throughout the pentagon — diagonal-to-side ratio, apothem-to-side ratio, and inscribed pentagram all contain $\varphi$.
The Greek school of Pythagoras used the pentagram (five-pointed star inside a pentagon) as a secret symbol — partly because of the golden-ratio relationships inside it.
A Worked Example
Find the interior angle of a regular pentagon.
The intuitive (wrong) approach. A student divides $360°$ by 5 to get $72°$.
Why it fails. $360°$ is the sum of exterior angles, not interior. The exterior angle of a regular pentagon is $72°$ (one-fifth of $360°$), but the interior angle is different.
The correct method. Sum of interior angles $= (n - 2) \times 180° = 3 \times 180° = 540°$. For a regular pentagon, each interior angle $= 540° / 5 = 108°$.
Check. Interior + exterior at each vertex = $180°$ (linear pair). $108° + 72° = 180°$ ✓.
What Are the Most Common Mistakes With Pentagons?
Mistake 1: Using triangle area formulas directly
The fix: A pentagon isn't a triangle. Use the regular pentagon formula or divide into triangles for irregular ones.
Mistake 2: Confusing interior and exterior angles
The fix: Interior angle = $108°$ (regular pentagon). Exterior angle = $72°$. They're supplementary at each vertex.
Mistake 3: Forgetting that the formula assumes "regular"
The fix: The $A = 1.72 s^2$ formula is for a regular pentagon. For irregular pentagons, divide into triangles.
Key Takeaways
A pentagon has 5 sides, 5 vertices, and interior angles summing to $540°$.
A regular pentagon has all sides equal and all angles $= 108°$.
Area of a regular pentagon: $A \approx 1.72 s^2$ (or exactly $\tfrac{1}{4}\sqrt{5(5+2\sqrt{5})} \cdot s^2$).
Perimeter of a regular pentagon: $P = 5s$.
The golden ratio $\varphi$ appears throughout the regular pentagon — diagonal-to-side ratio is exactly $\varphi$.
A Practical Next Step
Try these three before moving on to hexagons and higher polygons.
Find the perimeter of a regular pentagon with side $12$ cm.
Find the area of a regular pentagon with side $8$ m.
Find the exterior angle of a regular pentagon.
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