Properties of Regular Polygons: Pentagon, Hexagon, Octagon and Decagon

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 9 Mathematics Textbook, Unit 6, Chapter 6.4.

~8 min read

Summary

An n-sided regular polygon has n lines of symmetry.
Every regular polygon has a circumcircle through all its vertices and an incircle that touches all its sides. The radius of the incircle is the apothem.
The central angle of an n-sided regular polygon is 360° ÷ n.
For radius r: side s = 2r sin(180°/n), apothem a = r cos(180°/n), perimeter P = ns, and area A = (1/2)aP.

Key Words

Line of symmetry: a line that divides a figure into two halves that match exactly, like a mirror.
Center of a regular polygon: the point inside the polygon where all its lines of symmetry meet.
Circumcircle: the circle that passes through all the vertices of the polygon. Its radius r runs from the center to a vertex.
Incircle: the circle inside the polygon that touches each side at its midpoint.
Apothem (a): the perpendicular segment from the center to a side. It is also the radius of the incircle.
Central angle: the angle at the center between two radii drawn to consecutive vertices.
Inscribed polygon: a polygon whose vertices all lie on a circle. The polygon is inscribed in the circle, and the circle is circumscribed about the polygon.

Lines of Symmetry

A figure is symmetric when one half is the exact mirror image of the other half. The folding line is called a line of symmetry.
All regular polygons are symmetric, and the rule is simple: an n-sided regular polygon has exactly n lines of symmetry.
An equilateral triangle has 3 lines, a square has 4, a regular pentagon has 5, and a regular hexagon has 6.
When n is odd, each line of symmetry runs from a vertex to the midpoint of the opposite side.
When n is even, half the lines join opposite vertices and the other half join the midpoints of opposite sides.
All the lines of symmetry cross at one point inside the polygon. That point is the center of the polygon.

The Circumcircle and the Incircle

For every regular polygon, you can draw a circle through all its vertices. This outer circle is the circumcircle, and its radius r goes from the center to any vertex.
You can also draw a smaller circle inside the polygon that touches each side at its midpoint. This inner circle is the incircle.
The radius of the incircle is the apothem of the polygon. It is the perpendicular distance from the center to a side.
We say the polygon is inscribed in the circumcircle, and the incircle is inscribed in the polygon.
Not every polygon has both circles, but every regular polygon does. This is one of the special properties that makes regular polygons useful.

The Central Angle

Draw radii from the center to two consecutive vertices. The angle between them at the center is a central angle.
The n radii cut the polygon into n identical isosceles triangles, and the angles around the center add up to 360°.
So the measure of each central angle of an n-sided regular polygon is 360° ÷ n.
Notice that this is the same number as each exterior angle from Chapter 6.3. For a pentagon both are 72°, and for an octagon both are 45°.

Worked example: find the central angle of a regular polygon with 5 sides and with 8 sides.

For n = 5, the central angle is 360° ÷ 5 = 72°.

For n = 8, the central angle is 360° ÷ 8 = 45°.

Side, Apothem, Perimeter and Area

Each of the n isosceles triangles has two sides equal to the radius r, and the apothem cuts its central angle in half. Half of 360°/n is 180°/n.
Trigonometry in that half triangle gives the four formulas of Theorem 6.1, listed below for an n-sided regular polygon inscribed in a circle of radius r.
Side: s = 2r sin(180°/n). This means half a side is r sin(180°/n).
Apothem: a = r cos(180°/n). The apothem is always a little shorter than the radius.
Perimeter: P = ns = 2nr sin(180°/n). The perimeter is just n copies of the side.
Area: A = (1/2)aP. Each triangle has area (1/2)sa, and there are n of them, so A = (1/2)a × ns = (1/2)aP.

The hexagon shortcut: in a regular hexagon the central angle is 60°, so each of the 6 triangles is equilateral. This gives s = r, P = 6r, a = (√3/2)r, and A = (3√3/2)r². Since the side equals the radius, you can also write the area as A = (3√3/2)s².

Worked example: find the side of a square inscribed in a circle of radius 5 cm.

Use s = 2r sin(180°/n) with r = 5 and n = 4.

s = 2 × 5 × sin(45°) = 10 × 1/√2 = 5√2 cm.

Worked example: find the apothem of a regular pentagon inscribed in a circle of radius 6 cm.

Use a = r cos(180°/n) with r = 6 and n = 5.

a = 6 × cos(36°) ≈ 6 × 0.8090 ≈ 4.854 cm.

Worked example: find the area of a regular nonagon inscribed in a circle of radius 10 cm.

Combine the formulas: A = (1/2)aP = nr² cos(180°/n) sin(180°/n).

With n = 9 and r = 10: A = 9 × 100 × cos(20°) × sin(20°).

Using the table values cos(20°) ≈ 0.9397 and sin(20°) ≈ 0.3420, we get A ≈ 289.24 cm².

The Four Polygons of This Chapter

Regular polygon	Sides (n)	Lines of symmetry	Central angle	Each interior angle
Pentagon	5	5	72°	108°
Hexagon	6	6	60°	120°
Octagon	8	8	45°	135°
Decagon	10	10	36°	144°

Common Mistakes to Avoid

Mixing up the radius and the apothem. The radius reaches a vertex, while the apothem reaches the midpoint of a side. The apothem is always shorter than the radius.
Using 360°/n in the side and apothem formulas. The formulas use the half angle 180°/n, because the apothem cuts the central angle in half.
Confusing the central angle with the interior angle. The central angle sits at the center and equals 360° ÷ n. The interior angle sits at a vertex and equals (n − 2) × 180° ÷ n.
Treating s = r as a general rule. The side equals the radius only for the regular hexagon, because only there the central triangles are equilateral.
Forgetting the 1/2 in the area formula A = (1/2)aP.

Easy Ways to Remember

Think “r to the corner, a to the side”: the radius touches a vertex and the apothem touches the middle of a side.
Three angles share one value: the central angle and the exterior angle of a regular polygon are both 360° ÷ n.
The pair sin and s go together: the side uses sine, and the apothem uses cosine.
One area formula rules them all: A = (1/2)aP works for every regular polygon, from the triangle to the decagon.
Symmetry is free information: n sides always means n lines of symmetry.

Quiz

Tap an answer to check it.

1. How many lines of symmetry does a regular octagon have?

2. What is the measure of each central angle of a 10-sided regular polygon?

3. The central angle of a regular polygon measures 12°. How many sides does the polygon have?

4. Each side of a regular hexagon is 10 units long. What is the area of this hexagon?

5. A square is inscribed in a circle of radius 4 cm. What is the length of each side of the square?

Remember: An n-sided regular polygon has n lines of symmetry, a circumcircle of radius r through its vertices, an incircle of radius a touching its sides, and a central angle of 360° ÷ n. Its measurements come from four formulas: s = 2r sin(180°/n), a = r cos(180°/n), P = ns, and A = (1/2)aP.