By definition, π is the ratio of a circle's circumference to its diameter, or of a disk's area to the square of its radius. It therefore seems only natural that it should appear in many formulas for areas and volumes: the surface area of a sphere, the lateral surface area of a cylinder or cone, the volume of a ball, cylinder or cone… But it can also be used to determine the area of an ellipse from the lengths of its axes, as well as the volume of an ellipsoid. This is hardly surprising, since ellipses are conic sections and can therefore be defined as central projections of a circle onto a plane.
It also crops up in many plane-geometry problems involving triangles, since the three angles of a triangle add up to π radians. Its use in trigonometry also leads to the many arctangent formulas used to calculate its decimal digits (see "The never-ending quest for decimal digits").
In physics
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The constant π appears in many formulas in physics, reflecting their geometric origins. For example:
• the period T of a simple pendulum of length l in a gravitational field g is given by
