Calculate the perimeter (circumference) and area of any common shape: circle, rectangle, triangle, square, regular polygon, or ellipse. Includes shape comparison and isoperimetric ratio.
The perimeter of a shape is the total length of its boundary — the distance you would walk if you traced around the outside edge. For a circle, the perimeter is called the circumference. Perimeter is one of the most fundamental measurements in geometry, essential for fencing, framing, border layouts, trim calculations, and any task that involves measuring "around" an object.
For simple shapes, the formulas are well-known: a circle's circumference is 2πr, a rectangle's perimeter is 2(l + w), and a square's is simply 4s. Triangles require adding all three sides. Regular polygons generalize to P = n × s, where n is the number of sides and s is the side length. The ellipse is trickier — there is no closed-form formula, but Ramanujan's approximation π(a + b)(1 + 3h/(10 + √(4 − 3h))) is accurate to within 0.04% for most practical cases.
Beyond raw perimeter, the isoperimetric ratio 4πA/P² measures how efficiently a shape encloses area relative to its boundary length. A circle achieves the maximum ratio of 1 — it encloses the most area for a given perimeter. All other shapes score less than 1. This ratio is used in image processing, biology (cell shape analysis), and urban planning (lot efficiency).
This calculator supports six shape types with automatic area computation, a perimeter-to-area ratio, circularity percentage, and a reference table comparing formulas across all supported shapes. Presets for common real-world objects (rooms, coins, fields) let you explore instantly.
Perimeter — Circle, Rectangle, Triangle, Polygon & Ellipse problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter radius, side length, length, and it returns perimeter, area, shape, p/a ratio in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Circle: C = 2πr, A = πr² Rectangle: P = 2(l + w), A = l × w Triangle: P = a + b + c, A = √(s(s−a)(s−b)(s−c)) (Heron's) Square: P = 4s, A = s² Regular n-gon: P = ns, A = (ns²/4)cot(π/n) Ellipse: C ≈ π(a+b)(1 + 3h/(10+√(4−3h))), A = πab Isoperimetric ratio: 4πA / P²
Result: Perimeter = 32 cm, Area = 60 cm², Circularity ≈ 73.6%
For a 10 × 6 rectangle: P = 2(10 + 6) = 32 cm. A = 10 × 6 = 60 cm². Isoperimetric ratio = 4π(60)/32² ≈ 0.736, meaning it encloses about 73.6% as efficiently as a circle of the same perimeter.
This perimeter — circle, rectangle, triangle, polygon & ellipse tool links the entered values (radius, side length, length, width) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Perimeter — Circle, Rectangle, Triangle, Polygon & Ellipse shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (perimeter, area, shape, p/a ratio) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
Perimeter is the total length of the boundary of a 2D shape. For a circle, it is called circumference.
Add the length and width, then multiply by 2: P = 2(l + w). For a 10 × 6 rectangle, P = 2(16) = 32.
The circumference C = 2πr, where r is the radius. For r = 5, C = 2π(5) ≈ 31.42.
The isoperimetric ratio 4πA/P² measures how efficiently a shape encloses area. It equals 1 for a circle and is less than 1 for all other shapes.
This calculator uses Ramanujan's second approximation, which is accurate to within about 0.04% for eccentricities up to 0.95 — more than sufficient for practical use.
P = n × s, where n is the number of sides and s is the side length. For a regular hexagon with side 5: P = 6 × 5 = 30.