Area Calculator

Area: A complete guide to shapes and formulas

Area measures the two-dimensional size of a shape. Whether you’re estimating flooring, land, material usage, or graphics, area calculations help you plan, compare, and optimize. This guide summarizes core formulas for common shapes and offers best practices for reliable results.

Core formulas

• Rectangle/Square: \(A=\text{length}\times\text{width}\).
• Triangle (base & height): \(A=\frac{1}{2}\times \text{base}\times \text{height}\).
• Triangle (Heron): \(A=\sqrt{s(s-a)(s-b)(s-c)}\), \(s=\frac{a+b+c}{2}\).
• Circle: \(A=\pi r^2\).
• Sector: \(A=\frac{\theta}{360^\circ}\cdot \pi r^2\).
• Trapezoid: \(A=\frac{(b_1+b_2)}{2}\cdot h\).
• Regular polygon (n sides): \(A=\frac{n\,s^2}{4\tan(\pi/n)}\).
• Custom polygon: Shoelace formula \(\displaystyle A=\frac{1}{2}\left|\sum x_i y_{i+1} - \sum y_i x_{i+1}\right|\).

Best practices

• Keep units consistent: Convert all dimensions to the same unit before calculating.
• Measure carefully: Small errors in length can cause large area differences.
• Validate angles: For polygons and sectors, check degrees are within feasible ranges.
• Document inputs: Record dimensions, units, and formulas in reports or quotes.

Worked examples