Area of Regular Polygons (The Apothem)
A regular polygon's area equals half the perimeter times the apothem — the perpendicular distance from the center to a side.
Introduction
Regular hexagon: a = apothem, r = radius, s = side
Past Knowledge
Triangle area (11.1.1). Regular polygons (9.1.1). Trig ratios (8.2).
Today's Goal
Define apothem and apply A = ½aP.
Future Success
Composite figures (11.1.4), 3D solids (12.2), as n→∞ this formula → πr².
Key Concepts
Area of a Regular Polygon
where = apothem (center ⊥ to side), = perimeter =
Finding the Apothem with Trig
Central angle = . The apothem bisects this angle, creating a right triangle:
Formula Derivation
A regular n-gon splits into congruent isosceles triangles, each with base and height :
∎ Sum of n triangle areas gives ½ × apothem × perimeter.
Worked Examples
Given a and P
Regular hexagon: apothem = 5√3, side = 10.
P = 6(10) = 60.
Finding Apothem with Trig
Regular pentagon with side 8. Find the area.
A ≈ 110.1
Common Pitfalls
Apothem ≠ Radius
The apothem goes center → midpoint of side (⊥). The radius goes center → vertex. The apothem is always shorter.
Forgetting to Multiply by n
P = ns, not just s. Make sure you use the full perimeter.
Real-Life Applications
Hexagonal Floor Tiles
Regular hexagons tessellate perfectly. Knowing the area of one tile (from its apothem) lets you calculate how many tiles cover a floor.
Stop Signs
Stop signs are regular octagons. The apothem formula calculates the area of reflective material needed for each sign.
Practice Quiz
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