Area Calculator
The area calculator gives the area of a circle, rectangle, triangle, parallelogram, trapezoid, ellipse or rhombus, with the perimeter and the formula for each.
Area
- Area
- 78.539816 square units
- Formula
- A = pi r^2 = pi x 5^2
- Circumference
- 31.415927
- If the units were feet
- 78.539816 sq ft = 8.7266463 sq yd = 0.0018030261 acres
- Doubling every length
- 314.15927 square units
How the Area Calculator works
Area is a count of unit squares, and every formula in this calculator is a way of doing that count without actually laying out the squares. The rectangle is the base case and is not really a formula at all: a rectangle w by h holds exactly w times h unit squares, which is what area means. Everything else is that result, rearranged.
A parallelogram is the clearest example. Cut the triangle off one end, slide it to the other, and you have a rectangle of the same base and the same perpendicular height. Nothing was added or removed, so the area is base times height. The slanted side never appears, because leaning a shape over does not change how much of the plane it covers. A triangle follows immediately: two copies of any triangle, one rotated half a turn, fit into a parallelogram, so a triangle is half base times height. The height in both is the perpendicular distance, not the length of a sloping edge, and confusing the two is the most common mistake in school geometry.
The trapezoid averages. Its two parallel sides differ, so take their mean and treat the shape as a rectangle of that width: area is (a + b)/2 times h. The same doubling trick proves it: two trapezoids, one inverted, make a parallelogram with base a + b.
The circle is the one that needs a limit rather than a rearrangement. Cut the disk into many thin wedges and lay them alternately point-up and point-down. The result approaches a rectangle whose height is the radius and whose width is half the circumference, pi r. Multiply: pi r squared. The formula is still the rectangle's, reached by taking the wedges thinner and thinner. An ellipse is then a stretched circle, and stretching by a factor multiplies area by that factor, so pi r squared becomes pi a b.
A rhombus is counted by its diagonals. They cross at right angles, so the rhombus fills exactly half the rectangle they span: d1 times d2, over 2.
Units square with the measurement: if the inputs are in feet, the area is in square feet.
Area Calculator FAQ
What is the formula for the area of a circle?
A = pi r squared, where r is the radius, half the diameter. If you have the diameter, the formula is pi d squared over 4. The most common error is using the diameter where the radius belongs, which makes the answer four times too big.
What is the area of a rectangle?
Width times height. It is the definition of area rather than a derived result: a rectangle holds exactly that many unit squares. If the sides are in different units, convert before multiplying, not after.
Why is a triangle's area half base times height?
Because two copies of any triangle, one rotated half a turn, fit exactly into a parallelogram with the same base and height. The parallelogram is base times height, so the triangle is half. The height must be the perpendicular distance from the base to the opposite vertex, not the length of a slanted side.
What is the area of a parallelogram?
Base times perpendicular height, the same as a rectangle with those dimensions. Slicing the triangle off one end and sliding it to the other turns one into the other without changing the area. The length of the slanted side is irrelevant.
How do I convert square feet to square yards?
Divide by 9, not by 3. A yard is 3 feet, so a square yard is 3 by 3 feet. Area conversions square the length factor, and getting this wrong by using 3 is the classic flooring-order mistake. It triples the order.
If I double the size of a shape, what happens to the area?
It quadruples, for every shape. Area scales with the square of any linear change, so doubling gives 4 times, tripling gives 9 times, and halving gives a quarter. A 16-inch pizza has 78% more area than a 12-inch one.
Why is there no simple formula for an ellipse's perimeter?
Because none exists in elementary functions. The exact value requires an elliptic integral, a function named after exactly this problem. Ramanujan's approximation, used here, is accurate to better than one part in ten million for any ordinary shape.
Area by shape
- Area of a parallelogram calculator
- Area of a trapezoid calculator
- Area of a square calculator
- Area of a rhombus calculator