Area overview
Area describes the size of a two-dimensional region and can be imagined as the amount of paint required to cover a surface. It is the planar counterpart to linear distance and three-dimensional volume.
The International System of Units (SI) uses the square meter (m^2) as the base unit of area. When complex shapes arise, they are commonly decomposed into simple figures, their areas are calculated, and the parts are summed.
The calculators above let you jump between shapes instantly, with matching narratives so every formula, unit, and story from the classic reference is present inside this modern studio.
Rectangle
A rectangle is a quadrilateral with four right angles. The longer dimension is typically called length, while the shorter dimension is width. When both dimensions match, the rectangle becomes a square.
Its area is the simplest one in planar geometry: area = length * width. Measure or enter the two perpendicular edges and multiply.
The Farmer and his Daughter - Unsold Land: a farmer fences a perfectly rectangular plot that measures 220 ft by 99 ft. Converting nothing, he computes 220 * 99 = 21,780 square feet, which equals half an acre. Foreign investors with different shoe sizes argue about units, the fence remains up, and the land sits unsold.
Triangle
Numerous formulas exist for the area of a triangle. In general situations, Heron (Hero) of Alexandria provided the most practical version: let s = (a + b + c)/2 be the semi-perimeter, then area = sqrt[s(s - a)(s - b)(s - c)].
When the triangle is right, you may fall back to 0.5 * base * height. If only two sides and the included angle are known, use 0.5 * a * b * sin(C); all of these paths are summarized inside the solver above.
The Farmer and his Daughter - Triangle Daze: after selling the rectangular field, the farmer splurges on a pool. His seven-year-old, obsessed with triangles after watching Dora explore Egypt, demands a 77 ft equilateral triangle. The semi-perimeter becomes 115.5 ft, the area evaluates to roughly 2,567.33 square feet, and the pool edges double as lap lanes that are half the length of an Olympic pool.
Need more than area? Launch the Triangle Calculator to solve every side, angle, height, and radius instantly.
Trapezoid
A trapezoid is a simple convex quadrilateral with at least one pair of parallel sides. Those sides are labeled b1 and b2, while the perpendicular distance between them is the height h.
Its area is the average of the bases multiplied by the height: area = ((b1 + b2)/2) * h. This works no matter the remaining angles as long as the figure stays simple and convex.
The Farmer and his Daughter - Ramping Endeavors: two years pass, the daughter gets into freestyle BMX, and a triangular-only phase evolves into an appreciation of shapes that still involve triangles. The ramp face becomes a trapezoid with h = 9 ft, b1 = 9 m (29.528 ft), and b2 = 9 ft. Plugging the numbers yields ((9 + 29.528)/2) * 9 = 173.376 square feet of plywood to construct.
Circle
A circle contains every point that sits a fixed distance (the radius) from a central point. That constant distance governs every value, including area.
The universal relation is area = pi * r^2. Once the radius is known, the calculation is immediate, and the calculator above reports the result along with unit conversions.
The Farmer and his Daughter - Circle of Li(f)es: at age 15, the daughter plans a prank involving crop circles to unsettle her superstitious father. She plots concentric rings with an outer radius of 15 ft, computes pi * 15^2 = 706.858 square feet, and executes the design overnight, inviting a crowd of cereologists to trample the fields.
Sector
A sector is a proportional slice of a full circle defined by two radii and the intercepted arc. When the central angle is measured in degrees, area = (theta / 360) * pi * r^2, and when the angle is in radians, area = 0.5 * r^2 * theta.
To apply the formula, convert the input angle to the chosen unit, determine the ratio between the slice and the entire circle, and multiply by the area of the parent circle.
The Farmer and his Daughter - Sectioning Family: the family raccoon Platypus devours half of a blackberry pie of radius 16 in. The remaining half must feed three people, so each gets a 60 degree sector. The baker computes (60/360) * pi * 16^2 = 134.041 square inches per person, while the raccoon watches with a full belly.
Ellipse
An ellipse is the locus of points for which the sum of distances to two foci is constant. The longest radius is called the semi-major axis a, and the shortest is the semi-minor axis b.
The area parallels that of a circle but uses both axes: area = pi * a * b. If a equals b, the ellipse collapses to a circle.
The Farmer and his Daughter - Falling out of Orbit: at 18, the daughter aims for elite universities, misses, and ends up building a model of Earth's near-elliptical orbit in her bedroom. With a = 20 ft and b = 18 ft, the area of her symbolic orbit becomes pi * 20 * 18 = 1,130.97 square feet.
Parallelogram
A parallelogram features two pairs of parallel, equal-length sides. Opposite angles match, and the shape can be rearranged into a rectangle, proving that its area equals base times perpendicular height.
Take any side as the base b, drop a perpendicular to obtain height h, and compute area = b * h. Rectangles, rhombuses, and squares are all special cases of this definition.
The Farmer and his Daughter - Diamond in the Sky: after more years of metaphorical space travel, an enormous octahedral diamond supposedly lands on the family farm. Measuring one rhomboidal face with base 20 ft and height 18 ft yields an area of 360 square feet. The daughter eventually sells the gemstone, trading symbolism for luxurious reality.