How to Calculate Area of a Polygon: Simple Step-by-Step Guide

Calculating the area of a polygon depends on its type and the information available. Regular polygons—those with equal sides and angles—can be solved using specific formulas. Irregular polygons require decomposition or coordinate-based methods. This guide covers reliable techniques for finding polygon area, with a clear example and practical takeaways.

Regular Polygons: Use the Apothem and Perimeter

For regular polygons, the area is calculated using the apothem (the perpendicular distance from the center to a side) and the perimeter (the sum of all side lengths). The formula is:

Area = (1/2) × apothem × perimeter

This works for any regular polygon, from triangles to polygons with many sides. First, compute the perimeter by multiplying the number of sides by the side length. If the apothem is not given, it can be found using trigonometry.

For example, consider a regular hexagon with side length 6 units. It has 6 sides, so the perimeter is 6 × 6 = 36 units. If the apothem is 5.2 units, the area is:

(1/2) × 5.2 × 36 = 93.6 square units.

This method is efficient when the apothem is known or can be measured.

Finding the Apothem Using Trigonometry

When the apothem is not provided, divide the polygon into isosceles triangles from the center to each vertex. Each central angle equals 360° divided by the number of sides. Then, split one triangle into two right triangles.

For a regular pentagon with side length 8 units:

• Central angle = 360° ÷ 5 = 72°
• Half of that angle = 36°
• Half the side length = 4 units

In the right triangle, tan(36°) = opposite / adjacent = 4 / apothem
So, apothem = 4 / tan(36°)

Using a calculator, tan(36°) ≈ 0.7265
Apothem ≈ 4 / 0.7265 ≈ 5.51 units

Perimeter = 5 × 8 = 40 units
Area = (1/2) × 5.51 × 40 ≈ 110.2 square units

This approach is consistent and works for any regular polygon. More details are available at Khan Academy.

Irregular Polygons: Break Into Triangles

Irregular polygons lack uniform sides and angles, so standard formulas do not apply. The most effective method is triangulation: divide the polygon into non-overlapping triangles, calculate each area, and sum them.

Choose one vertex and draw diagonals to non-adjacent vertices, forming triangles. Use the triangle area formula:

Area = (1/2) × base × height

If height is not available but two sides and the included angle are known, use:

Area = (1/2) × a × b × sin©

For example, take a quadrilateral with vertices A, B, C, and D. Draw diagonal AC to form triangles ABC and ADC.

• Triangle ABC: base = 10 units, height = 4 units → area = (1/2) × 10 × 4 = 20
• Triangle ADC: base = 10 units, height = 3 units → area = (1/2) × 10 × 3 = 15

Total area = 20 + 15 = 35 square units

This method works for convex and concave polygons alike, as long as the shape is simple (no intersecting sides). It is widely used in geometry and design. A clear explanation can be found at Math is Fun.

Coordinate Geometry: Use the Shoelace Formula

When vertex coordinates are known, the shoelace formula computes area directly. List the vertices in order—clockwise or counterclockwise—and repeat the first point at the end.

For vertices (x₁,y₁), (x₂,y₂), …, (xₙ,yₙ), calculate:

• Sum1 = x₁y₂ + x₂y₃ + … + xₙy₁
• Sum2 = y₁x₂ + y₂x₃ + … + yₙx₁
• Area = (1/2) × |Sum1 − Sum2|

Example: Triangle with vertices (1,1), (4,5), (7,2)

List points: (1,1), (4,5), (7,2), (1,1)

Sum1 = (1×5) + (4×2) + (7×1) = 5 + 8 + 7 = 20
Sum2 = (1×4) + (5×7) + (2×1) = 4 + 35 + 2 = 41
Area = (1/2) × |20 − 41| = (1/2) × 21 = 10.5 square units

The shoelace formula is accurate and widely used in surveying, computer graphics, and mapping. It avoids estimation and works for any simple polygon. A visual walkthrough is available at Math is Fun.

Special Cases and Common Formulas

Some polygons have simple, well-established area formulas:

• Triangle: (1/2) × base × height
  (Or Heron’s formula if all three sides are known)
• Rectangle: length × width
• Parallelogram: base × height (height must be perpendicular to base)
• Trapezoid: (1/2) × (sum of parallel sides) × height
• Regular polygon (alternative): (n × s²) / (4 × tan(180°/n)), where n is the number of sides and s is side length

For a regular octagon with side length 5:

• n = 8, s = 5
• Area = (8 × 5²) / (4 × tan(22.5°)) = (8 × 25) / (4 × tan(22.5°))
• tan(22.5°) ≈ 0.4142 → denominator ≈ 4 × 0.4142 = 1.6568
• Area ≈ 200 / 1.6568 ≈ 120.7 square units

This formula is derived from the apothem method but combines steps. It is useful when only side length and number of sides are known. More on polygon geometry is available at Khan Academy.

Key Takeaways

• For regular polygons, use Area = (1/2) × apothem × perimeter. If the apothem is unknown, calculate it using trigonometry based on the central angle and half the side length.
• For irregular polygons, divide the shape into triangles and sum their areas. This method works with physical drawings or geometric models.
• When vertex coordinates are given, apply the shoelace formula: list points in order, compute two cross-sums, and take half the absolute difference.
• Use standard formulas for common shapes—rectangle, triangle, trapezoid—whenever applicable to save time and reduce error.
• Always ensure consistent units, and verify that your calculator is in the correct angle mode (degrees or radians) when using trigonometric functions.