Pentagon Shape - Properties, Area, and Perimeter
Pentagon Shape - Properties, Area, and Perimeter
TL;DR
A pentagon is a polygon with 5 sides and 5 interior angles summing to 540°. A regular pentagon has all sides equal and all angles equal to 108° each. Its area formula is A=(\frac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \cdot s^2\approx 1.72 s^2), and its perimeter is P=5s.
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)×180°=(5−2)×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 = \frac{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 \text{ or } A\approx 1.72s^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} ]
Irregular Pentagon
No single formula. Divide into 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=5×7=35 \text{ cm} ]
Standard — Area Using Side
A regular pentagon has side 6 m. Find its area. [ A\approx 1.72×6^2\approx 61.94 \text{ m}^2 ]
Stretch — Area Using Apothem
A regular pentagon has side 4 cm and apothem 2.75 cm. Find its area. [ A=\frac{5\cdot4\cdot2.75}{2}=27.5 \text{ cm}^2 ]
Why Does the Pentagon Matter? (The Real-World GROUND)
Pentagons appear in nature, architecture, and science:
- The Pentagon (US Department of Defense headquarters) is a regular-pentagon-shaped building.
- Sea stars (starfish) have 5-fold radial symmetry.
- Flowers — many flower species have 5 petals.
- Pentaprisms — used in cameras and surveying instruments.
- Crystalline structures — pentagonal symmetry found in quasicrystals.
- Soccer balls — have pentagonal panels.
A Worked Example
Find the interior angle of a regular pentagon. The correct method. Sum of interior angles: ( (n−2)×180°=(3×180°)=540° ) For a regular pentagon, each interior angle =( 540°/5=108° ).
What Are the Most Common Mistakes With Pentagons?
Mistake 1: Using triangle area formulas directly
The fix: Use the pentagon formula or divide into triangles for irregular ones.
Mistake 2: Confusing interior and exterior angles
The fix: Interior angle = 108°. Exterior angle = 72°.
Mistake 3: Forgetting that the formula assumes "regular"
The fix: The area formula is for a regular pentagon.
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≈1.72s².
- Perimeter of a regular pentagon: P=5s.
- The golden ratio (\varphi) appears throughout the regular pentagon.