Area Calculator
Below you’ll find a set of calculators designed to compute the surface of seven frequently‑encountered geometric figures. When faced with a more intricate outline, you can usually split it into simpler components, evaluate each part, and sum the results. These tools are especially handy for approximating the size of a parcel of land.
Rectangle
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Triangle
Use the Triangle Calculator to determine |
Trapezoid
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Circle
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Sector
 |
Ellipse
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Parallelogram
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Area measures how much two‑dimensional space a figure occupies. Imagine the amount of paint needed to coat a flat surface – that’s essentially what area tells you. It serves as the planar counterpart to a line’s length or a solid’s volume. In the SI system the standard unit is the square metre ( m2 ). Below are the formulas for several basic shapes together with sample calculations.
Rectangle
A rectangle is a four‑sided polygon whose interior angles are all right angles. Determining its area is straightforward once you know its length and width. By definition a quadrilateral has four edges and four vertices; for a rectangle the longer edges are called the length and the shorter ones the width. When both dimensions match the shape becomes a square, a special case of a rectangle. The area is obtained with the following expression:
area = length × width
The Farmer and his Daughter – Unsold Land
Picture a farmer who wants to put his perfectly rectangular field on the market. He fenced it to keep the cattle in and measured each side. Living in the U.S., he still uses feet rather than metres. Since 1959 the foot has been fixed at exactly 0.3048 m, a legacy of using the human body as a length reference, which varied over time and locale. The plot measures 220 ft in length and 99 ft in width. With these numbers you can compute its area:
area = 220 × 99 = 21780 sq ft
The field’s surface works out to 21,780 sq ft, which is roughly half an acre (an acre historically being defined as a chain by a furlong, among other old measures). Unfortunately the farmer’s region attracts investors who prefer smaller foot‑based measurements and expect more square footage for their money, leaving the lot unsold.
Triangle
There are numerous ways to find a triangle’s area depending on the data you have. For detailed guidance, head over to the Triangle Calculator. The method illustrated above is based on Heron’s formula (occasionally credited to Hero of Alexandria), a classic technique named after the ancient Greek mathematician and engineer. The formula is presented below:
area = √s(s - x)(s - y)(s - z)
| where: |
|
|||||
| x, y, z are the sides of the triangle |
The Farmer and his Daughter - Triangle Daze
After a great deal of hard work the farmer finally sold his 21,780 sq ft parcel and earmarked part of the proceeds for a family pool. He neglected to consider that a year’s worth of pool upkeep could easily fund a season’s worth of water‑park visits for his kids. Adding to the drama, his 7‑year‑old daughter—who’s become enamoured with triangles after a virtual trip to Egypt via Dora the Explorer—demands a triangular pool whose dimensions involve the number 7 to mark her age. Yielding to her wishes, the farmer opts for an equilateral triangle with each side measuring 77 ft, believing this size will fit his backyard.
| s = |
|
= 115.5 |
area = √115.5 × (115.5 - 77)3 = 2567.33 sq ft
Since the longest distance between any two points of an equilateral triangle is the length of the edge of the triangle, the farmer reserves the edges of the pool for swimming "laps" in his triangular pool with a maximum length approximately half that of an Olympic pool, but with double the area – all under the watchful eyes of the presiding queen of the pool, his daughter, and the disapproving glare of his wife.
Trapezoid
A trapezoid is a convex quadrilateral possessing at least one pair of parallel sides. Convexity guarantees that none of its interior angles exceed 180°, and the shape does not intersect itself. The parallel sides are called the bases, while the non‑parallel sides are the legs. Although trapezoids come in several varieties, their area is always calculated with the same formula:
| area = |
|
× h |
The Farmer and his Daughter – Ramping Endeavors
Two years after the pool was installed, the farmer’s daughter has grown older and her interests have broadened. Though she still loves triangles, she now recognises that geometry alone can’t keep the world turning, and even Santa’s workshop wouldn’t balance on a pyramid‑shaped North Pole. Embracing other shapes, she’s taken up freestyle BMX and needs a skate ramp. The catch? The ramp must be built from pieces that can be assembled using only triangles, and every measurement must feature the number 9 to reflect her age. Comfortable with metric units, the farmer designs a ramp whose side profile is a trapezoid 9 ft tall, with a bottom base of 29.528 ft (≈9 m) and a top base of 9 ft. The trapezoid’s area is worked out as follows:
| area = |
|
× 9 = 173.376 sq ft |
Circle
A circle constitutes a closed curve consisting of all planar points that lie at a constant distance from a fixed centre. That constant distance is known as the radius. Further information about circles is available on the Circle Calculator page; however, determining its area only requires the radius and the relationship governed by the constant π. The formula for a circle's area is:
area = πr2
The Farmer and his Daughter – Circle of Li(f)es
Six years later the girl is a confident 15-year-old who spends most of her time chasing likes on social media, often neglecting the support of family. After a clash with her dad over her online habits, she exploits his superstition to play a prank, gathering advice from strangers about UFOs and crop circles. Convinced that the Earth is spherical and abandoning her previous posts, she now wants to design a simple crop‑circle pattern made of concentric circles, needing the area for a circle whose outer radius measures 15 ft. She applies the formula below:
area = π × 152 = 706.858 sq ft
Unfortunately for the farmer, not only is he terrified of the crop circle that appeared overnight on the night that his daughter told him she was at a slumber party with her friends, that for some odd reason did not result in superfluous Instagram posts (he was of course his daughter's first follower), but the number of "circle investigators" and "cereologists" showing up on his farm to examine, and subsequently confirm the authenticity of the crop circle as an alien construction, cost him significant damages to his crops.
Sector
A circular sector represents the slice of a disk bounded by two radii and the connecting arc. If you know the radius and the central angle, the sector's area equals the full circle's area multiplied by the fraction of the angle over 360° (or 2π radians), as expressed in the formula below:
| area = |
|
× πr2 | if θ is in degrees |
or
| area = |
|
× πr2 | if θ is in radians |
The Farmer and his Daughter – Sectioning Family
The family now faces a fresh dilemma. A year after her birthday, the 16-year-old's mother baked a blackberry pie, her favorite treat. Their pet raccoon, Platypus, has already devoured a 180° wedge, leaving crumbs that lead straight to him. Previously the pie could be shared among three people and the raccoon, but now only half the pie remains for the three guests while a content Platypus watches. Each guest will receive a 60° slice from a pie with a 16‑inch radius, and the area of each portion is computed as follows:
area= 60°/360° × π × 162 = 134.041 in2
As a result of Platypus' inconsideration, each person gets one-third less pie, and the daughter contemplatively recalls American history class, where she learned about the Battle of the Alamo and the portrayal of the folk hero Davy Crockett and his coonskin hat.
Ellipse
An ellipse is the generalized form of a circle, and is a curve in a plane where the sum of the distances from any point on the curve to each of its two focal points is constant, as shown in the figure below, where P is any point on the ellipse, and F1 and F2 are the two foci.
When the two focal distances are equal (F₁ = F₂), the ellipse collapses into a perfect circle. In the illustration accompanying the calculator, the longest radius of an ellipse is the semi‑major axis, while the shortest is the semi‑minor axis. These axes correspond to diameters, not radii. The area of an ellipse is calculated similarly to that of a circle, differing only by the presence of two distinct radii (since the foci coincide for a circle):
area = πab
where a and b are the semi-major
and semi-minor axes
The Farmer and his Daughter – Falling out of Orbit
Two years after the disappearance of their raccoon Platypus, the daughter wins a furry accessory in a school raffle, easing the loss. Now 18, she plans to leave rural Montana for a college experience filled with freedom and indulgence, along with some studies. Raised in an environment of constant encouragement, she internalized the mantra "shoot for the moon; even if you miss, you'll land among the stars," and she was told she could achieve anything. However, with mediocre grades, no extracurriculars—her varied interests consume all her free time—no planning, and an insistence on applying only to elite universities, the rejection she received felt like being flung into deep space: inflating, freezing, and suffocating as she missed the moon and crashed among the stars. Along with her disappointment, her aspiration to become an astrophysicist was temporarily crushed, and she turned to calculating the elliptical area needed to construct a life‑size model of Earth's near‑elliptical orbit around the sun inside her bedroom, allowing her to gaze at a central sun representing her passion, surrounded by the cold vacuum of space, with Earth's distant rotation symbolizing the gap between her dreams and reality.
area = π × 18 ft × 20 ft= 1130.97 sq ft
Parallelogram
A parallelogram is a simple quadrilateral featuring two pairs of parallel sides, with opposite sides and angles equal in length and measure. Rectangles, rhombuses, and squares are special cases of parallelograms. The term "simple" indicates that the shape does not intersect itself. By dissecting a parallelogram into a right triangle and a trapezoid, then rearranging them, you can form a rectangle, which leads to the same area formula used for rectangles. Instead of length and width, a parallelogram uses base and height—the height being the perpendicular distance between the bases. Based on the diagram below, the area formula for a parallelogram is:
area = b × h
The Farmer and his Daughter – Diamond in the Sky
Two more years pass for the farmer's family, and although his daughter's actions once caused serious concern, she finally reconciles the blazing sun of her heart with the Earth that society expects her to stay grounded on. After enduring self‑imposed isolation and imagined judgment from all sides, she emerges like a diamond, radiant and resolute. Recognizing life's challenges, she chooses to persevere through the asteroid field of existence, hoping for a fairy‑tale ending. At last, hope appears—not as a prince, but as a sign from the heavens. In her poetic musings, she imagines that her influence caused a massive octahedral diamond‑shaped asteroid to land safely on their farm, symbolising her journey and rebirth. She now measures the area of one rhomboidal face of this newfound emblem:
area = 20 ft × 18 ft = 360 sq ft
Unfortunately for the farmer's daughter, the appearance of the enormous diamond drew attention from all over the world, and after sufficient pressure, she succumbs to the human within her, and sells the diamond, the very representation of her life and soul, to a wealthy collector, and proceeds to live the rest of her life in lavish indulgence, abandoning her convictions, and losing herself within the black hole of society.
Common Area Units
| Unit | Area in m2 |
| square meter | SI Unit |
| hectare | 10,000 |
| square kilometre (km2) | 1,000,000 |
| square foot | 0.0929 |
| square yard | 0.8361 |
| acre | 4,046.9 (43,560 square feet) |
| square mile | 2,589,988 (640 acres) |