Geometry
Perimeter and area calculator
Pick the figure, enter the measurements and get perimeter and area together with the derived quantities: diagonal, apothem, height, interior angle.
How perimeter and area are used
Perimeter measures the outline, area the surface it encloses: two different questions that almost always arrive together. A garden fence needs the perimeter, the lawn to be seeded needs the area, and the same figure can have a large perimeter and a small area — a long, narrow rectangle shows that at once.
The triangle here is solved from its three sides with Heron's formula: take the semi-perimeter s = (a+b+c)/2 and the area is the square root of s(s−a)(s−b)(s−c). That matches what a tape measure can actually give you, whereas the height nearly always has to be derived. For completeness the calculator also returns the height on the base, which is what the school formula asks for.
The regular polygon joins the two worlds: perimeter and area follow from the number of sides and the side length, by way of the apothem — the distance from the centre to the midpoint of a side. As the number of sides grows the polygon approaches the circle, and the two results converge.
Common mistakes
- Mixing up radius and diameter on a circle: halve the diameter before entering it, or the area comes out four times too large.
- Using the slanted side instead of the height on a parallelogram or trapezoid: the height is the perpendicular distance between the bases, always less than or equal to the side.
- Mixing units: if one measurement is in metres and another in centimetres the result means nothing. Convert first, then calculate.
Frequently asked questions
Can I find the area of a triangle without knowing the height?
Yes, and it is the usual case: the three sides are enough. Heron's formula returns the area, and the calculator also shows the height on the base as a derived measurement.
Why is the perimeter of an ellipse approximate?
Because no elementary formula exists: the exact perimeter is an elliptic integral, which is where the name comes from. Ramanujan's approximation used here is off by less than one part in a billion for any reasonable ellipse — far less than the error in your measurement of the axes.
What unit does the result come in?
The same one you used for the inputs. Measure in metres and the perimeter is in metres, the area in square metres: the calculator applies no conversion.
How this calculation works
Square: P = 4l, A = l². Rectangle: P = 2(b+h), A = b·h. Triangle (Heron): s = (a+b+c)/2, A = √(s(s−a)(s−b)(s−c)). Circle: C = 2πr, A = πr². Parallelogram: P = 2(b+l), A = b·h. Rhombus: P = 4l, A = (d₁·d₂)/2. Trapezoid: P = B+b+l₁+l₂, A = (B+b)·h/2. Regular polygon with n sides: P = n·l, apothem = l/(2·tan(π/n)), A = P·apothem/2. Ellipse: A = π·a·b, perimeter ≈ π(a+b)[1 + 3h/(10+√(4−3h))] with h = (a−b)²/(a+b)², Ramanujan's approximation.
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