How do I calculate the area of a regular polygon?

Area = (1/2) × perimeter × apothem = (1/2) × n × s × a, where n = number of sides, s = side length, and a = apothem (distance from center to midpoint of a side). Alternatively: Area = (n × s² × cot(π/n)) / 4. For specific regular polygons: equilateral triangle (n=3): A = (√3/4)s², square (n=4): A = s², pentagon (n=5): A ≈ 1.72s², hexagon (n=6): A = (3√3/2)s² ≈ 2.60s², octagon (n=8): A = 2(1+√2)s² ≈ 4.83s². The apothem can be calculated as a = s/(2·tan(π/n)). As the number of sides increases, a regular polygon approaches a circle — a 100-gon is virtually indistinguishable from a circle, and the area formula converges to πr².

What are the interior and exterior angle formulas for polygons?

Interior angle sum = (n-2) × 180°, where n = number of sides. Each interior angle of a regular polygon = (n-2) × 180° / n. Exterior angle sum is always 360° for any convex polygon. Each exterior angle of a regular polygon = 360° / n. Examples: triangle (n=3): interior 60° each, exterior 120°. Square (n=4): 90°, 90°. Pentagon (n=5): 108°, 72°. Hexagon (n=6): 120°, 60°. Octagon (n=8): 135°, 45°. Decagon (n=10): 144°, 36°. Practical application: regular hexagons tile a plane perfectly (interior angle 120° divides evenly into 360°), which is why honeycombs, hex floor tiles, and game boards use hexagonal grids. Only regular triangles, squares, and hexagons can tile by themselves without gaps.

What is the formula for the sum of the interior angles of any polygon?

The sum of the interior angles of any n-sided polygon is given by the formula (n - 2) × 180°. For example, a quadrilateral (n=4) has a sum of (4 - 2) × 180° = 360°, while a hexagon (n=6) has a sum of (6 - 2) × 180° = 720°. This formula applies to both regular and irregular convex polygons.

How many diagonals can be drawn in a polygon?

The number of diagonals, D, in an n-sided polygon is calculated using the formula D = n(n - 3) / 2. For instance, a square (n=4) has D = 4(4 - 3) / 2 = 2 diagonals. A pentagon (n=5) has D = 5(5 - 3) / 2 = 5 diagonals, connecting non-adjacent vertices.

How is the perimeter of a regular polygon determined?

The perimeter of a regular polygon is the total length of its boundary. It is calculated by multiplying the length of one side (s) by the number of sides (n), using the formula P = n × s. For example, a regular octagon with sides measuring 5 cm each has a perimeter of 8 × 5 cm = 40 cm.

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Number of sides (n)

Side Length (s)

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n= number of sidess= side lengtha= apothem (center to midpoint of side)A= areaP= perimeter

Perimeter

Total length of all sides.

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Interior angle

Each interior angle of a regular polygon.

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Area (from side length)

Using the apothem derived from the side.

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Area (from apothem)

Half the perimeter times the apothem.

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Apothem from side length

Perpendicular distance from center to a side.

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n= number of sides

s= side length

a= apothem (center to midpoint of side)

A= area

P= perimeter

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