Area Calculator
PrintArea is the quantitative measurement describing the extent or size of a two-dimensional surface in a plane. You can visualize it as the total amount of paint required to cover a flat surface. Area serves as the two-dimensional counterpart to one-dimensional curve lengths and three-dimensional solid volumes. The standard unit of area under the International System of Units (SI) is the square meter (m²), though real-world applications frequently require conversions to square feet, acres, or hectares.
Our free Area Calculator solves the surface area of 7 common planar shapes instantly. By selecting your shape and entering its dimensions, you can estimate land areas, compute building layouts, and compare metric and imperial units.
Area Formulas and Practical Examples
Each shape relies on a distinct geometric formula to evaluate its two-dimensional space:
1. Rectangle
A rectangle is a four-sided polygon (quadrilateral) with four right angles. If all four sides are equal, it forms a square.
Area = length × width
The Unsold Land: A farmer in the United States wants to sell a fenced rectangular plot of land measuring 220 feet in length and 99 feet in width.
Area = 220 × 99 = 21,780 sq ft (equivalent to exactly 0.5 acres).
2. Triangle (Heron’s Formula)
Heron’s formula solves a triangle’s area using its three side lengths (x, y, z) without needing the height:
Area = √[ s(s - x)(s - y)(s - z) ]
Where the semiperimeter (s) is: s = (x + y + z) / 2
The Triangular Pool: To satisfy his daughter’s love of pyramids, a father builds an equilateral pool in his backyard with side lengths of 77 ft:
s = (77 + 77 + 77) / 2 = 115.5
Area = √[ 115.5 × (115.5 - 77)³ ] ≈ 2,567.33 sq ft
3. Trapezoid
A simple convex quadrilateral with at least one pair of parallel sides (bases, b1 and b2).
Area = [ (b1 + b2) / 2 ] × h (where h is the vertical height between the bases).
The BMX Ramp: A father builds a ramp for his daughter where the trapezoidal side face features a height of 9 ft, a top base of 9 ft, and a bottom base of 29.528 ft (9 meters):
Area = [ (9 + 29.528) / 2 ] × 9 = 173.376 sq ft
4. Circle
A closed curve where all points are equidistant from a center point, related through the mathematical constant π.
Area = π × r² (where r is the radius).
The Crop Circle: To prank her father, a teenager maps out a circular pattern in a wheat field with a radius of 15 ft:
Area = π × 15² ≈ 706.858 sq ft
5. Circular Sector
A wedge-shaped slice of a circle enclosed by two radii and a boundary arc.
- In Degrees:
Area = (A / 360) × π × r² - In Radians:
Area = (θ / 2π) × π × r² = ½ × r² × θ
The Missing Pie: A pet raccoon eats half of a circular blackberry pie (180°). The remaining 180° is split equally among three family members (60° each). If the pie has a radius of 16 inches:
Area = (60 / 360) × π × 16² ≈ 134.041 in²
6. Ellipse
A elongated closed curve with two focal points, defined by a semi-major axis (longer radius, a) and a semi-minor axis (shorter radius, b).
Area = π × a × b
The Orbit Model: An astronomy student builds a scale model of the Earth’s orbit in her room, using axes of 18 ft and 20 ft:
Area = π × 18 × 20 ≈ 1,130.97 sq ft
7. Parallelogram
A quadrilateral with two pairs of parallel sides. It can be restructured into a rectangle by moving a triangular section from one side to the other.
Area = base × height
The Meteorite Asteroid: A massive diamond asteroid lands on a farm. The daughter measures one of the rhomboidal parallel faces, which has a base of 20 ft and a height of 18 ft:
Area = 20 × 18 = 360 sq ft
Common Area Unit Conversions
Convert your calculated areas using these standard metric and imperial unit conversion values:
| Area Unit | Equivalent in Square Meters (m²) | Equivalent in US Customary Units |
|---|---|---|
| Square Meter (m²) | 1 m² | 10.764 sq ft |
| Hectare (ha) | 10,000 m² | 2.471 acres |
| Square Kilometer (km²) | 1,000,000 m² | 247.105 acres |
| Square Foot (ft²) | 0.0929 m² | 144 sq inches |
| Square Yard (yd²) | 0.8361 m² | 9 sq feet |
| Acre (ac) | 4,046.9 m² | 43,560 sq feet |
| Square Mile (mi²) | 2,589,988 m² | 640 acres |
Determine three-dimensional capacities on our Volume Calculator or solve right-angle triangle boundaries using the Triangle Calculator.
Frequently Asked Questions (FAQ)
What is the difference between area and surface area?
Area refers to the two-dimensional space occupied by a flat, planar shape (like a drawing on a piece of paper). Surface area describes the total boundary area of a three-dimensional solid object (like the outer skin of a sphere, cube, or cylinder exposed to the air).
How do you calculate the area of complex, irregular shapes?
To calculate the area of an irregular boundary, the most common approach is decomposition. You break down the complex figure into several smaller, standard shapes (such as triangles and rectangles), calculate the individual areas of those shapes, and sum them together. For curved irregular shapes, integral calculus is required.
Why is an acre defined as 43,560 square feet?
The acre is an old imperial unit historically defined as the amount of land that could be plowed by a yoke of oxen in a single day. Geometrically, it was defined as a strip of land one chain wide (66 feet) by one furlong long (660 feet). Multiplying these dimensions yields 43,560 square feet.
How does an ellipse’s area formula relate to a circle?
The formula for an ellipse is Area = π × a × b, where a and b are the semi-major and semi-minor axes (radii). If the ellipse changes shape so that the major and minor axes become equal (a = b = r), the shape becomes a perfect circle, and the formula simplifies back to Area = π × r².