Pentagon Shape - Properties, Area, and Perimeter

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.

1. Find the perimeter of a regular pentagon with side $12$ cm.

2. Find the area of a regular pentagon with side $8$ m.

3. Find the exterior angle of a regular pentagon.

Also Read:

• Geometry — Concepts, Formulas, Shapes & Topics Guide

• Geometric Shapes

• Shapes in Geometry

• Hexagon Shape

• Interior Angles

• Parallelogram

• Rhombus