The short answer
Area is length × width for a rectangle, π × radius² for a circle, and ½ × base × height for a triangle — every other shape on this page builds on those three formulas. Pick your shape, enter its dimensions in one consistent unit, and the calculator returns the area, perimeter, the formula used, and a step-by-step breakdown.
Key takeaways
- Area is always expressed in square units (ft², m², cm²) while perimeter is a single linear measurement (ft, m, cm) — mixing the two up is the most common geometry mistake.
- For a circle, the formula uses radius, not diameter. Entering a diameter into a radius field roughly quadruples the calculated area.
- Regular polygon formulas (pentagon, hexagon, octagon) only work for shapes with equal sides and angles. Irregular polygons need to be split into triangles or rectangles first.
- Keep every dimension in the same unit before calculating — mixing feet and inches in the same shape produces an area that's off by a large, inconsistent factor.
Area formulas for every shape
| Shape | Formula | Example |
|---|---|---|
| Rectangle | A = l × w | 5 m × 3 m = 15 m² |
| Square | A = s² | 4 ft side = 16 ft² |
| Circle | A = π × r² | r = 6 ft ≈ 113.10 ft² |
| Triangle | A = ½ × b × h | b = 8 ft, h = 5 ft = 20 ft² |
| Trapezoid | A = ½ × (b₁ + b₂) × h | b₁ = 6, b₂ = 4, h = 3 = 15 units² |
| Parallelogram | A = b × h | b = 10, h = 4 = 40 units² |
| Rhombus | A = ½ × d₁ × d₂ | d₁ = 8, d₂ = 6 = 24 units² |
| Ellipse | A = π × a × b | a = 5, b = 3 ≈ 47.12 units² |
| Regular hexagon | A ≈ 2.598 × s² | s = 4 ≈ 41.57 units² |
| Sector | A = (θ/360) × π × r² | r = 10, θ = 90° ≈ 78.54 units² |
| Annulus (ring) | A = π × (R² − r²) | R = 5, r = 3 ≈ 50.27 units² |
Pentagon and octagon follow the same pattern as hexagon — area ≈ constant × side², where the constant comes from the regular-polygon formula n × s² ÷ (4 × tan(π/n)). This calculator applies the exact formula for whichever shape you select, so results stay precise rather than relying on rounded constants.
Worked example: area of a circle
A = π × r²
A = π × 6² = π × 36 ≈ 113.10 ft²
For a circle with a 6 ft radius, square the radius first (6² = 36), then multiply by π (≈3.14159) to get roughly 113.10 square feet. The circumference — the distance around the circle — uses a different formula, 2 × π × r, which gives about 37.70 ft for the same circle. Area and circumference both start from the radius but answer different questions: how much surface is inside the circle versus how far you'd walk around its edge.
Common mistakes to avoid
- Using diameter instead of radius. If you only know the diameter, divide it by 2 before entering it into a radius field — otherwise the area comes out roughly 4x too large.
- Mixing units within one shape. A trapezoid with bases measured in feet and a height measured in inches will not produce a valid area until every dimension shares the same unit.
- Treating irregular shapes as regular ones. The hexagon and octagon formulas assume equal sides and angles — an irregular six-sided lot needs to be broken into triangles or rectangles and summed instead.
- Reporting area in linear units. "20 feet" is a length; the correct area unit is "20 square feet" (20 ft²) — dropping the exponent is a common transcription error on homework and permits.
Related calculators
- Circle Calculator — solve for radius, diameter, circumference, or area from any one known value.
- Triangle Calculator — find sides, angles, and area from partial triangle information.
- Surface Area Calculator — extend area into 3D for spheres, cylinders, cones, and boxes.
- Volume Calculator — calculate the space enclosed by a 3D solid rather than a flat shape.