Calculate the area of any 2D shape: circle, rectangle, triangle, trapezoid, ellipse, and more.
About the Area Calculator
The Area Calculator is a precise, free online math tool built for students, teachers, engineers, and professionals. It handles complex calculations instantly using floating-point arithmetic accurate to multiple decimal places — no installation, no account, no cost.
Area is a physical quantity describing the size or planar extent of a two-dimensional figure. It can be visualized as the total amount of paint required to cover a flat surface. This page outlines standard formulas, measurement units, and practical geometric examples.
This tool is part of our comprehensive Math Calculators collection, working alongside our Percentage Calculator, Fraction Calculator, and Statistics Calculator.
📐 Core Formulas & Methodology
Rectangle: A = l × w | Circle: A = πr² | Triangle: A = ½bh or Heron's Formula
Area Calculations for Common 2D Shapes
Each geometric shape requires specific parameters to determine its exact area. Below is an analytical look at the shapes supported by this calculator.
1. Rectangle
A rectangle is a quadrilateral featuring four right angles. Calculating its area simply requires knowing its length and width. When length and width are identical, the shape represents a square.
Formula: Area = length × width
Imagine a farmer selling a rectangular plot of land in the United States. He measures his fenced plot in traditional feet: a length of 220 feet and a width of 99 feet.
Calculation: 220 × 99 = 21,780 sq ft (exactly half an acre). Because foreign investors in his region favor metric measurements, his plot remains on the market.
2. Triangle (Heron’s Formula)
While the simple base-height formula is common, this calculator implements Heron’s formula to compute area when only side lengths are known. For advanced cases, utilize our dedicated Triangle Calculator or Right Triangle Calculator.
Formula: Area = √[s(s - x)(s - y)(s - z)] Where semiperimeter (s) = (x + y + z) / 2
The farmer decides to build an equilateral pool for his daughter, who insists that every side must be exactly 77 feet to represent her age of 7.
s = (77 + 77 + 77) / 2 = 115.5
Area = √[115.5 × (115.5 – 77)³] ≈ 2,567.33 sq ft.
3. Trapezoid
A trapezoid is a simple convex quadrilateral featuring at least one pair of parallel sides (called bases). The other two sides are known as the legs.
Formula: Area = [(b1 + b2) / 2] × h Where b1, b2 are the parallel bases, and h is vertical height
Two years later, the daughter pursues freestyle BMX and requests a non-curved ramp with a trapezoidal side profile. Using a height of 9 feet, a bottom base of 29.528 feet (9 meters), and a top base of 9 feet:
Area = [(9 + 29.528) / 2] × 9 = 173.376 sq ft.
4. Circle
A circle represents a closed planar shape where all points reside at an equal distance (the radius) from a central locus. Try our dedicated Circle Calculator for precise diameter, circumference, and radius evaluations.
Formula: Area = πr²
At age 15, the daughter plays a crop-circle prank on her father, structuring concentric circles in their wheat field. The outermost boundary has a radius of 15 feet.
Area = π × 15² ≈ 706.858 sq ft.
The subsequent influx of investigators examining the “alien formation” causes notable crop damage.
5. Sector
A sector is a portion of a circle enclosed by two radii and an arc, resembling a slice of pie.
Formula: Area = (θ / 360) × πr² (if θ is in degrees) Formula: Area = (θ / 2π) × πr² (if θ is in radians)
On the daughter’s 16th birthday, their pet raccoon eats half (180 degrees) of her favorite blackberry pie. The remaining half of the pie, with a radius of 16 inches, is split equally among three family members (60 degrees each).
Area = (60 / 360) × π × 16² ≈ 134.041 sq in.
6. Ellipse
An ellipse is a generalized oval-like form of a circle where the sum of the distances from any point on the curve to its two focal points remains constant.
Formula: Area = π × a × b Where a and b represent the semi-major and semi-minor axes (radii)
At age 18, facing college rejection letters, the daughter constructs an elliptical orbital projection in her room to visualize space mechanics. It features semi-axes of 18 feet and 20 feet:
Area = π × 18 × 20 ≈ 1,130.97 sq ft.
7. Parallelogram
A parallelogram is a simple quadrilateral featuring two pairs of parallel sides. Because it can be mathematically rearranged into a rectangle, its area calculation follows a similar base-height structure.
Formula: Area = base × height
At age 20, a massive octahedral asteroid composed of diamond lands safely in their pasture. The daughter measures one of its flat, rhomboidal parallelogram faces, having a base of 20 feet and height of 18 feet:
Area = 20 × 18 = 360 sq ft.
Standard Area Unit Conversions
The SI standard unit for area is the square meter (m²). The table below lists common imperial and metric conversions to help with scale transformations:
| Unit | Value in Square Meters (m²) | Alternative Equivalence |
|---|---|---|
| Square Meter (m²) | 1.0 | SI base unit |
| Hectare (ha) | 10,000 | 100 m × 100 m |
| Square Kilometer (km²) | 1,000,000 | 100 hectares |
| Square Foot (sq ft) | 0.0929 | 144 square inches |
| Square Yard (sq yd) | 0.8361 | 9 square feet |
| Acre (ac) | 4,046.9 | 43,560 square feet |
| Square Mile (sq mi) | 2,589,988 | 640 acres |
How to Use the Area Calculator
Select your target 2D shape, input the dimensions in the respective numeric fields, and click Calculate. Results update instantly with high precision. Adjusting any active input automatically refreshes the area calculations in real time.
🎓 Educational Value
These geometric equations align directly with core school mathematics curricula worldwide. Utilizing the tool can help students verify their homework, construct accurate physical models, and visualize advanced spatial concepts.
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📐 Precision & Accuracy Notes
- Calculated outputs display up to 8 significant figures; trailing decimal zeros are cropped to maintain a readable layout.
- Highly polarized measurements (extremely large or small areas) default to scientific notation formats.
- Ensure all measurements use identical unit systems before entering values to prevent calculations from skewing.
- Utilize our dedicated Scientific Calculator for advanced multi-variable algebraic or trigonometric equations.
📚 Authoritative Resources