Regular Polygons SAT Notes

Regular-polygon formulas become easier when the center is connected to every vertex. The figure breaks into congruent central triangles whose angle and area relationships explain the formulas.

Regular and inscribed polygons

Regular polygon: A convex polygon with all sides congruent and all interior angles congruent.

Inscribed polygon: Every polygon vertex lies on one circle; that circle is circumscribed about the polygon.

Regular hexagon inscribed in a circleA regular hexagon has all six vertices on one circle. Center O, radii to consecutive vertices, and an apothem perpendicular to one side are shown.

Radius versus apothem

Radius R

Center to a vertex of the circumscribed circle.

Apothem a

Perpendicular distance from center to a side; it meets that side at its midpoint.

Radius, central angle, and apothemA regular pentagon shows center O, radius R to a vertex, apothem a to the midpoint of a side, a right-angle mark, and one 72-degree central angle.

Angle formulas

Regular-polygon angle formulas
Quantity	Formula	Applies to
Sum of interior angles	$(n-2)180^\circ$	Any n-gon
Each interior angle	$\frac{(n-2)180^\circ}{n}$	Regular n-gon
Sum of exterior angles	$360^\circ$	Any polygon
Each exterior angle	$360^\circ/n$	Regular n-gon
Central angle	$360^\circ/n$	Regular inscribed n-gon

Worked example

Find n from an angle

A regular polygon has each exterior angle equal to $24^\circ$ . Find n and each interior angle.

Sides
$$n=360/24=15$$ .
Interior
$180-24=156^\circ$ .

15 sides; each interior angle is

156^\circ

Familiar regular polygons

Familiar regular-polygon angles
Polygon	n	Central angle	Interior angle
Equilateral triangle	3	120°	60°
Square	4	90°	90°
Regular pentagon	5	72°	108°
Regular hexagon	6	60°	120°
Regular octagon	8	45°	135°

Area from central triangles

Regular-polygon area decompositionA regular hexagon is divided by radii into six congruent central triangles. One radius and the apothem to a side are labeled.

Regular-polygon area

\[A=\frac12ap\]

The polygon is n congruent triangles, each with base s and height a: $n(\tfrac12sa)=\tfrac12a(ns)=\tfrac12ap$ .

Area workflow

Perimeter
Compute $$p=ns$$ when side length is known.
Apothem
Use the half-central right triangle if a is not given.
Area
Substitute full perimeter and apothem into $A=\tfrac12ap$ .
Check
Verify the apothem is perpendicular to a side, not drawn to a vertex.

Worked example

Regular hexagon from side length

A regular hexagon has side length 8. Find its exact area.

Central triangle
The 60-degree central triangle splits into two 30-60-90 triangles. Half-side is 4, so $a=4\sqrt3$ .
Perimeter
$$p=6(8)=48$$ .
Area
$A=\tfrac12(4\sqrt3)(48)=96\sqrt3$ .

96\sqrt3

square units.

Apothem from radius

\[a=R\cos\left(\frac{180^\circ}{n}\right)\]

The apothem is adjacent to the half-central angle and the radius is the hypotenuse.

Worked example

Pentagon apothem from radius

A regular pentagon has radius 10. Express its apothem using cosine.

Half-central angle
$180^\circ/5=36^\circ$ .
Cosine
$\cos36^\circ=a/10$ .

a=10\cos36^\circ

Mini check

Angle and area check

A regular octagon has apothem 12 and side 10. Find its central angle, perimeter, and area.

Show answer and explanation

Answer: $45^\circ$ , 80, and 480 square units.

Use $$360/8$$ , $$p=8(10)$$ , then $A=\tfrac12(12)(80)$ .

Key takeaways

Central and regular exterior angles equal $360^\circ/n$ .
Interior sum is $(n-2)180^\circ$ ; divide by n only for each regular interior angle.
Radius ends at a vertex; apothem ends perpendicularly on a side.
Regular-polygon area uses apothem times full perimeter, then multiplies by one-half.

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Apply the ideas with SAT-style questions, then reinforce key details with flashcards.

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