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The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is: [6] A = 4πr 2 (sphere), where r is the radius of the sphere.
This is a list of volume formulas of basic shapes: [4]: 405–406 ... List of surface-area-to-volume ratios – Surface area per unit volume;
The area formula can be used in calculating the volume of a partially-filled cylindrical tank lying horizontally. In the design of windows or doors with rounded tops, c and h may be the only known values and can be used to calculate R for the draftsman's compass setting.
The surface-area-to-volume ratio has physical dimension inverse length (L −1) and is therefore expressed in units of inverse metre (m-1) or its prefixed unit multiples and submultiples. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm 2 and a volume of 1 cm 3. The surface to volume ratio for this cube is thus
The volume and area formulas may be derived by examining the rotation of the function = = for [,], using the formulas the surface of the rotation for the area and the solid of the revolution for the volume. The area is
Since the area of the rectangle is ab, the area of the ellipse is π ab/4. We can also consider analogous measurements in higher dimensions. For example, we may wish to find the volume inside a sphere. When we have a formula for the surface area, we can use the same kind of "onion" approach we used for the disk.
The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.
The volume and area formulas were first determined in Archimedes's On the Sphere and Cylinder by the method of exhaustion. Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.