What is a regular polygon?
A polygon is a closed flat shape made of straight sides. It is regular when two things are true: all sides are equal and all angles are equal. Both are needed. A rectangle has equal angles but not equal sides. A rhombus has equal sides but not equal angles. Neither is regular. A square is regular.
Names: equilateral triangle (3), square (4), regular pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), dodecagon (12).
Elements
- Vertex: a corner. Side a: an edge.
- Centre O: the point the same distance from every vertex (and from every side).
- Circumradius R: distance from O to a vertex.
- Apothem r (inradius): distance from O to the middle of a side. It is perpendicular to the side.
- Central angle: angle at O between two neighbouring vertices, 360° ÷ n.
- Interior angle: angle inside at a vertex, (n − 2) × 180° ÷ n. Exterior angle = 360° ÷ n.
Inscribed and circumscribed circles
Every regular polygon has two special circles with the same centre O:
- The circumscribed circle (circumcircle) passes through all vertices. Its radius is R. We say the polygon is inscribed in this circle.
- The inscribed circle (incircle) touches every side at its midpoint. Its radius is r. The polygon is circumscribed about this circle.
Why do they exist? Join O to all vertices: you get n identical isosceles triangles (two sides are R). Their heights from O are all equal to r. So all vertices are R from O, and all sides are r from O.
In one of these triangles, half a side, the apothem and R make a right-angled triangle with angle 180°/n at O. So R² = r² + (a/2)², and r is always less than R.
Formulas for R, r, side and area
From the right triangle with angle 180°/n at O:
- a = 2R sin(180°/n), so R = a ÷ (2 sin(180°/n))
- a = 2r tan(180°/n), so r = a ÷ (2 tan(180°/n))
- r = R cos(180°/n)
| Polygon | R | r | a in terms of R |
|---|---|---|---|
| Equilateral triangle | a/√3 | a/(2√3) | R√3 |
| Square | a/√2 | a/2 | R√2 |
| Regular hexagon | a | a√3/2 | R |
Note for the triangle: r = R/2.
Area
The polygon is n triangles, each with base a and height r. So Area = n × ½ × a × r = ½ × perimeter × r. For a hexagon this gives (3√3/2)a².
As n grows, the polygon hugs its circle: perimeter → 2πR and area → πR². This is how Archimedes estimated π using a 96-sided polygon.
Constructing a regular hexagon and a square
Hexagon (compass and straightedge)
- Draw a circle with centre O and radius R.
- Keep the compass at the same width R. Put the point anywhere on the circle and mark an arc on the circle.
- Move the point to the new mark and repeat. Six marks fit exactly, because the hexagon's side equals R.
- Join the marks in order. Join every second mark to get an equilateral triangle.
Square
- Draw a circle with centre O.
- Draw a diameter AC.
- Construct the perpendicular bisector of AC through O; it meets the circle at B and D.
- Join A, B, C, D. Bisecting the right angles again gives a regular octagon.
Try it
In the 3D free-play step, set n = 3, 4, 6 and then 12. Predict the interior angle before you look. Then draw a circle of radius 5 cm on paper and step the compass round it 6 times. Did the sixth mark land on the first? Measure a side: it should be 5 cm.
Key formulas and definitions
- Central angle = 360° ÷ n
- Interior angle = (n − 2) × 180° ÷ n; exterior angle = 360° ÷ n
- R = a ÷ (2 sin(180°/n)); r = a ÷ (2 tan(180°/n)); r = R cos(180°/n)
- R² = r² + (a/2)²
- Triangle: R = a/√3, r = a/(2√3); Square: R = a/√2, r = a/2; Hexagon: R = a, r = a√3/2
- Area = ½ × perimeter × r = ½ n a r
Worked examples
1. Find the interior angle of a regular octagon.
n = 8. Interior angle = (8 − 2) × 180° ÷ 8 = 1080° ÷ 8 = 135°.
2. Each interior angle of a regular polygon is 140°. How many sides does it have?
Exterior angle = 180° − 140° = 40°. n = 360° ÷ 40° = 9. It is a nonagon.
3. A square has side 4 cm. Find R and r.
r = a/2 = 2 cm. R = a/√2 = 4/1.414 ≈ 2.83 cm. Check: R² = 2² + 2² = 8, R = √8 ≈ 2.83 ✓.
4. An equilateral triangle has side 6 cm. Find R and r.
R = a/√3 = 6/1.732 ≈ 3.46 cm (= 2√3). r = a/(2√3) = 6/3.464 ≈ 1.73 cm (= √3). Note r = R/2.
5. A regular hexagon is inscribed in a circle of radius 5 cm. Find its side, apothem and area.
For a hexagon a = R = 5 cm. r = a√3/2 = 5 × 0.866 ≈ 4.33 cm. Area = ½ × perimeter × r = ½ × 30 × 4.33 ≈ 64.95 cm².
6. The apothem of a square is 3 cm. Find the radius of its circumscribed circle.
r = a/2, so a = 6 cm. R = a/√2 = 6/√2 = 3√2 ≈ 4.24 cm. (Or R = r ÷ cos 45° = 3 ÷ 0.707.)
7. A regular pentagon has side 10 cm. Find R and r (sin 36° ≈ 0.588, tan 36° ≈ 0.727).
180°/5 = 36°. R = 10 ÷ (2 × 0.588) ≈ 8.51 cm. r = 10 ÷ (2 × 0.727) ≈ 6.88 cm. Area = ½ × 50 × 6.88 ≈ 172 cm².
Common mistakes
- Calling a rhombus or rectangle regular. Regular needs BOTH equal sides and equal angles.
- Mixing up R and r: R goes to a corner, r goes to the middle of a side (and is shorter).
- Using 360°/n as the interior angle. That is the central (and exterior) angle; interior = 180° − 360°/n.
- Using sin(360°/n) in R = a/(2 sin …). The angle in the formula is HALF the central angle, 180°/n.