Area calculator
Area formulas by shape: rectangle is length × width, circle is πr², triangle is half base × height, trapezoid is half the sum of the parallel sides × height. The circle works from any one measurement, so a circumference of 31.42 gives a radius of 5 and an area of 78.54.
How to calculate area
The one that catches people out is the ellipse perimeter, which has no exact closed form at all: it needs an elliptic integral. Ramanujan published an approximation in 1914 that is accurate to better than one part in ten million for anything but the most extreme ellipse, and that is what is used here. It is a nice reminder that "simple shape" and "simple formula" are two different claims. The area of an ellipse is trivial and its perimeter is genuinely hard.
The circle takes whichever measurement you have
A circle is the shape most often approached from the wrong end, so the tab accepts any one of its four measurements as the input. A tape measure around a pipe or a tree gives you the circumference; dividing by 2π gives the radius, and everything else follows from there. A circumference of 31.42 comes back as a radius of 5, a diameter of 10 and an area of 78.54.
Doubling the radius doubles the circumference and quadruples the area, because one is linear in r and the other goes with r². That single fact explains why a 16-inch pizza is 1.78 times the pizza of a 12-inch, why a wider pipe carries disproportionately more water, and why doubling the diameter of a telescope collects four times the light. It is also why comparing round things by diameter alone consistently understates the difference.
Units, and the mistake they cause
The two kinds of output are not in the same units. Area comes out in the entered unit squared; circumference, radius and diameter come out in the unit itself. Mixing them is the most common arithmetic mistake on this page, and it shows up as an answer wrong by a whole factor of the unit instead of by a small amount. π itself is worth a word here too: 22/7 is good to about three decimal places, 3.1429 against 3.14159, which is fine for mental arithmetic and not for anything that has to be cut to size.
For a shape none of these tabs covers, split it into the ones they do. A room with a bay window is a rectangle plus a circle segment; a tapered garden bed is a trapezoid. Calculate each piece and add them, which is what a surveyor does by hand.
What people use it for
- Working out floor or wall area
- Sizing material for a curved or tapered shape
- Finding a radius or diameter from a measured circumference
- Comparing pizza or pipe sizes honestly
- Checking an area against a plan
- Geometry homework
Questions
π times the radius squared. A radius of 5 gives 78.54.
Half the base times the perpendicular height. For three known sides, use Heron formula in the triangle calculator.
2πr, or πd if you have the diameter. A radius of 5 gives 31.42.
Divide by 2π. A circumference of 31.42 gives a radius of 5.
π times the diameter squared, divided by four.
Half the sum of the two parallel sides, times the perpendicular height between them.
It has no exact elementary formula. Ramanujan approximation is used here and is accurate to about one part in ten million.
More than twice; 1.78 times the area, because area goes with the square of the diameter.
To about three decimal places: 3.1429 against 3.14159. Fine for mental arithmetic, and not for anything cut to size.
Split it into rectangles, triangles and circle segments, calculate each, and add them.
The circle area times the fraction of 360 degrees the angle covers. A 90-degree sector is a quarter of the circle; the sector tab takes the angle directly.
Because they measure different things. Enter centimetres and the area comes back in square centimetres while the perimeter stays in centimetres. Reading one as the other is the usual reason an answer is out by a whole factor.