Circle and Ellipse Area Calculator
Computes circle area from radius and ellipse area from semi-axes a and b using the standard pi formulas.
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Area of a circle and an ellipse
A circle is just the set of points that sit the same distance from a center, and its area works out to A = π·r². The ellipse is the more general shape: it has two semi-axes, a (the major) and b (the minor), so the area becomes A = π·a·b. Set a equal to b and you're back to a circle. How stretched the ellipse looks comes from its eccentricity, e = √(1 − b²/a²), which runs from 0 for a circle up toward 1 for something very elongated. Plug in r = 5 and you get A = 25π ≈ 78.540; with a = 4 and b = 3 the area is 12π ≈ 37.699.
Applications
Athletics tracks pair long straights with semicircular or elliptical ends, and you need their area to budget turf and surfacing. Planetary orbits trace ellipses, per Kepler's first law, while the area a planet sweeps out in a given time stays constant (that's the second law). The same geometry shows up in elliptical billiards and whispering-gallery acoustic chambers, where rays bounce from one focus to the other. Medical lithotripsy borrows the trick, focusing shock waves on a single point to break up kidney stones.
FAQ
Why is the area π·a·b and not π·(a+b)/2 squared? Picture a unit circle stretched by a along the x-axis and by b along the y-axis. That's an affine map, and area under it grows by the determinant a·b, so you land on π·a·b.
What is eccentricity used for? It puts a number on how stretched the ellipse is, which matters when you're working with orbital mechanics, optics or engineering tolerances.
Does this formula work in any unit? It does. Whatever linear unit you feed into r, a and b, the answer comes back squared (m → m², cm → cm²).
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Minimum Bend Radius
Estimate a sheet's minimum bend radius, R_min = t·(50/r − 1), from the thickness t (mm) and the material's percent reduction of area r in the tensile test (%, a ductility measure). The minimum radius is the smallest inner radius you can bend a sheet to WITHOUT cracking the outer face (which is in tension). Bending below the minimum causes cracks or rupture in the outer fiber, where tensile strain exceeds the material's capacity. The minimum radius depends strongly on the material's DUCTILITY (here via reduction of area r): very ductile materials (annealed aluminum, low-carbon steels) can be bent to nearly zero radius (sharp bend), while brittle or work-hardened materials need large radii. It also depends on the bend ORIENTATION relative to the sheet's rolling direction (bending across the rolling direction allows smaller radii than along it, due to anisotropy). Knowing the minimum radius is essential in bent-part design: specifying a smaller radius than possible leads to crack scrap. It is common to express the minimum radius as multiples of thickness (e.g. '2t'). Enter the thickness and the material's reduction of area.
Cam Pitch Radius
Calculate the pitch radius of a roller-follower cam, R_p = R_b + R_r, from the base circle radius R_b (mm) and the follower roller radius R_r (mm). In ROLLER-follower cams (a bearing rolling on the cam profile, reducing friction versus flat-face or knife-edge followers), two important curves are distinguished: the real PROFILE of the cam (the physical surface the roller touches) and the PITCH curve, the locus of the roller CENTER as it follows the cam. The pitch curve is designed first (from the displacement diagram), and the real profile is obtained by 'offsetting' the roller radius from the pitch curve. The pitch radius, at the base position, is the sum of the base circle radius and the roller radius. This distinction is fundamental for a practical reason: the roller radius cannot exceed the smallest RADIUS OF CURVATURE of the pitch curve in CONCAVE regions, or the roller does not 'fit' and the cam gets an incorrect profile (undercutting), distorting the motion. So the choice of roller radius and base radius is coupled to the cam geometry. The pitch radius also enters the pressure-angle and peripheral-speed calculations. Enter the base circle radius and the roller radius.
Ellipse Area & Perimeter
Compute area A = π·a·b and approximate perimeter (Ramanujan) of an ellipse from semi-axes a and b.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.