Geometric measurements

Surface area

Common formulas

Surface areas of common solids

Shape	Equation	Variables
Cube		s = side length
Cuboid		ℓ = length, w = width, h = height
Triangular prism		b = base length of triangle, h = height of triangle, l = distance between triangular bases, a, b, c = sides of triangle
All prisms		B = the area of one base, P = the perimeter of one base, h = height
Sphere		r = radius of sphere, d = diameter
Spherical lune		r = radius of sphere, θ = dihedral angle
Torus		r = minor radius (radius of the tube), R = major radius (distance from center of tube to center of torus)
Closed cylinder		r = radius of the circular base, h = height of the cylinder
Lateral surface area of a cone	................	............
......

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Volume

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i. e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

Video: Volume

The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.

Units

Volume measurements from the 1914 The New Student's Reference Work.

Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre (cm³) is the volume of a cube whose sides are one centimetre (1 cm) in length.

In the International System of Units (SI), the standard unit of volume is the cubic metre (m³). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus

1 litre = (10 cm)³ = 1000 cubic centimetres = 0.001 cubic metres,

so

1 cubic metre = 1000 litres.

Small amounts of liquid are often measured in millilitres, where

1 millilitre = 0.001 litres = 1 cubic centimetre.

In the same way, large amounts can be measured in megalitres, where

1 million litres = 1000 cubic metres = 1 megalitre. 

Volume formulas

Shape	Volume formula	Variables
Cube		a = length of any side (or edge)
Circular Cylinder		r = radius of circular base, h = height
Prism		B = area of the base, h = height
Cuboid		l = length, w = width, h = height
Triangular prism		b = base length of triangle, h = height of triangle, l = length of prism or distance between the triangular bases
Triangular prism (with given lengths of three sides)		a, b, and c = lengths of sides h = height of the triangular prism
Sphere		r = radius of sphere d = diameter of sphere which is the integral of the surface area of a sphere
Ellipsoid		a, b, c = semi-axes of ellipsoid
Torus		..........

.........

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Circumference

In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses. Informally, "circumference" may also refer to the edge itself rather than to the length of the edge.

Video: Circumference of a circle 

Circumference of a circle

Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circumference = π × diameter = 2 × π × radius.

The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this can not be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound.^[3] The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.

When a circle's diameter is 1, its circumference is π.

When a circle's radius is 1—called a unit circle—its circumference is 2π.

Relationship with π

The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter π. The first few decimal digits of the numerical value of π are 3.141592653589793 ...^[4] Pi is defined as the ratio of a circle's circumference C to its diameter d:

Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference:

The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science.

In Measurement of a Circle written circa 250 BCE, Archimedes showed that .......

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