Regular Polygon Practice Problems

Formulas you need

• Perimeter: $P = n s$
• Apothem: $a = \dfrac{s}{2\tan(180^\circ/n)}$
• Area: $A = \tfrac{1}{2} a P$
• Interior angle: $\dfrac{(n-2)\cdot 180^\circ}{n}$

where n = number of sides, s = side length, a = apothem (center to edge midpoint), P = perimeter

Generate a new problem

Pick a difficulty, work it out, then check your answer. Every press gives a new question — the numbers are drawn fresh each time, so you can keep going.

Warm-up

1. A regular polygon has 8 sides and side length s = 3. Find the perimeter (P).

   Answer & working

   24 units

   P = n · s = 8 · 3 = 24

2. A regular polygon has 8 sides and side length s = 1. Find the apothem (a).

   Answer & working

   1.21 units

   a = s / (2 tan(180°/n)) ≈ 1.21

3. A regular polygon has 4 sides and side length s = 1. Find the area (A).

   Answer & working

   1 units²

   A = ½ · a · P = ½ · 0.5 · 4 ≈ 1

4. A regular polygon has 8 sides. Find the interior angle (∠).

   Answer & working

   135 °

   interior ∠ = (n−2)·180°/n = 135°

5. A regular polygon has 3 sides and side length s = 3. Find the perimeter (P).

   Answer & working

   9 units

   P = n · s = 3 · 3 = 9

Core practice

1. A regular polygon has 10 sides and side length s = 1. Find the perimeter (P).

   Answer & working

   10 units

   P = n · s = 10 · 1 = 10

2. A regular polygon has 5 sides and side length s = 10. Find the apothem (a).

   Answer & working

   6.88 units

   a = s / (2 tan(180°/n)) ≈ 6.88

3. A regular polygon has 10 sides and side length s = 6. Find the area (A).

   Answer & working

   276.99 units²

   A = ½ · a · P = ½ · 9.2331 · 60 ≈ 276.99

4. A regular polygon has 6 sides. Find the interior angle (∠).

   Answer & working

   120 °

   interior ∠ = (n−2)·180°/n = 120°

5. A regular polygon has 12 sides and side length s = 10. Find the perimeter (P).

   Answer & working

   120 units

   P = n · s = 12 · 10 = 120

Stretch

1. A regular polygon has 4 sides and side length s = 2.5. Find the perimeter (P).

   Answer & working

   10 units

   P = n · s = 4 · 2.5 = 10

2. A regular polygon has 4 sides and side length s = 6.5. Find the apothem (a).

   Answer & working

   3.25 units

   a = s / (2 tan(180°/n)) ≈ 3.25

3. A regular polygon has 8 sides and side length s = 9.5. Find the area (A).

   Answer & working

   435.77 units²

   A = ½ · a · P = ½ · 11.4675 · 76 ≈ 435.77

4. A regular polygon has 9 sides. Find the interior angle (∠).

   Answer & working

   140 °

   interior ∠ = (n−2)·180°/n = 140°

5. A regular polygon has 3 sides and side length s = 6.5. Find the perimeter (P).

   Answer & working

   19.5 units

   P = n · s = 3 · 6.5 = 19.5