How many diagonals does a regular pentagon have?

A regular pentagon has 5 diagonals. You can find this using the diagonal formula for any polygon: $\frac{n(n-3)}{2} = \frac{5(5-3)}{2} = 5$. These five diagonals form a smaller regular pentagon inside, creating the famous pentagram (five-pointed star) pattern.

Regular Pentagon — Definition, Formula & Examples

A regular pentagon is a five-sided polygon where all five sides have the same length and all five interior angles are equal, each measuring 108°.

A regular pentagon is an equilateral and equiangular polygon with exactly five vertices, five edges of congruent length, and five congruent interior angles of $108°$ each. The sum of its interior angles is $540°$ , and it possesses five lines of symmetry with rotational symmetry of order 5.

Key Formula

A = \frac{1}{2} \times 5s \times a = \frac{5s^2}{4\tan(36°)}

Where:

$A$ = Area of the regular pentagon
$s$ = Length of one side
$a$ = Apothem (distance from center to midpoint of a side)

How It Works

To work with a regular pentagon, you typically need to know the side length

s

. From

s

, you can find the apothem (the distance from the center to the midpoint of a side), the perimeter, and the area. The perimeter is simply

5s

. The area formula uses the apothem:

A = \frac{1}{2} \times \text{perimeter} \times \text{apothem}

. You can also calculate the apothem directly from the side length using

a = \frac{s}{2\tan(36°)}

. Regular pentagons appear in nature (cross-sections of okra, starfish symmetry) and in design (the Pentagon building, soccer balls).

Worked Example

Problem: Find the area of a regular pentagon with a side length of 10 cm.

Step 1: Find the perimeter by multiplying the side length by 5.

P = 5 \times 10 = 50 \text{ cm}

Step 2: Calculate the apothem using the formula. Note that tan(36°) ≈ 0.7265.

a = \frac{s}{2\tan(36°)} = \frac{10}{2 \times 0.7265} \approx 6.88 \text{ cm}

Step 3: Apply the area formula using the perimeter and apothem.

A = \frac{1}{2} \times 50 \times 6.88 \approx 172.05 \text{ cm}^2

Answer: The area of the regular pentagon is approximately 172.05 cm².

Another Example

Problem: A regular pentagon has a perimeter of 35 cm. Find each interior angle and the length of one side.

Step 1: Find one side by dividing the perimeter by 5.

s = \frac{35}{5} = 7 \text{ cm}

Step 2: Each interior angle of a regular pentagon is found using the interior angle formula for regular polygons.

\text{Interior angle} = \frac{(n-2) \times 180°}{n} = \frac{(5-2) \times 180°}{5} = \frac{540°}{5} = 108°

Answer: Each side is 7 cm and each interior angle is 108°.

Visualization

Why It Matters

Regular pentagons come up frequently in middle-school and high-school geometry when studying polygon properties, interior angle sums, and area calculations. Architects and engineers use pentagonal geometry in structural design — the U.S. Department of Defense headquarters is literally named after this shape. Understanding the regular pentagon also builds the foundation for working with more complex regular polygons like hexagons and decagons.

Common Mistakes

Mistake: Assuming each interior angle is 100° or dividing 360° by 5 to get 72°.

Correction: The sum of interior angles is

(5-2) \times 180° = 540°

, so each angle is

540° \div 5 = 108°

. The value 72° is actually each exterior angle, not the interior angle.

Mistake: Confusing the apothem with the radius (distance from center to a vertex).

Correction: The apothem goes from the center to the midpoint of a side and is perpendicular to that side. The radius goes from the center to a vertex and is longer than the apothem.

Related Terms

Apothem — Used to calculate a regular pentagon's area
Diagonal of a Polygon — A regular pentagon has exactly 5 diagonals
Decagon — A regular polygon with twice as many sides
Heptagon — Another regular polygon with more sides
Dodecagon — A 12-sided regular polygon for comparison
Convex — All regular pentagons are convex polygons
Concave — Opposite of convex; a regular pentagon is never concave