In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having two and only two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.
The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings.
The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.
Formulas
Height
For any isosceles triangle, the following six line segments coincide:
the altitude, a line segment from the apex perpendicular to the base,
the angle bisector from the apex to the base,
the median from the apex to the midpoint of the base,
the perpendicular bisector of the base within the triangle,
the segment within the triangle of the unique axis of symmetry of the triangle, and
the segment within the triangle of the Euler line of the triangle.
Video: Height of Isosceles Triangle
Their common length is the height of the triangle. If the triangle has equal sides of length and base of length , the general triangle formulas for the lengths of these segments all simplify to
This formula can also be derived from the Pythagorean theorem using the fact that .....
A quartile is a type of quantile. The first quartile (Q1) is defined as the middle number between the smallest number and the median of the data set. The second quartile (Q2) is the median of the data. The third quartile (Q3) is the middle value between the median and the highest value of the data set.
In applications of statistics such as epidemiology, sociology and finance, the quartiles of a ranked set of data values are the four subsets whose boundaries are the three quartile points. Thus an individual item might be described as being "in the upper quartile".
Definitions
Boxplot (with quartiles and an interquartile range) and a probability density function (pdf) of a normal N(0,1σ2) population
Symbol
Names
Definition
Q1
first quartile
lower quartile
25th percentile
splits off the lowest 25% of data from the highest 75%
Q2
second quartile
median
50th percentile
cuts data set in half
Q3
third quartile
upper quartile
75th percentile
splits off the highest 25% of data from the lowest 75%
Computing methods
For discrete distributions, there is no universal agreement on selecting the quartile values.[1]
Method 1
Use the median to divide the ordered data set .....
The rhombus has a square as a special case, and is a special case of a kite and parallelogram.
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.
Every rhombus is a parallelogram and a kite. A rhombus with right angles is a square.
Characterizations
A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:
a parallelogram in which a diagonal bisects an interior angle
a parallelogram in which at least two consecutive sides are equal in length
a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
a quadrilateral with four sides of equal length (by definition)
a quadrilateral in which the diagonals are perpendicular and bisect each other
a quadrilateral in which each diagonal bisects two opposite interior angles
a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, and DAP are all congruent
a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point
Basic properties
Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties:
Opposite angles of a rhombus have equal measure.
The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral.
Its diagonals bisect opposite angles.
The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q, in every rhombus
Not every parallelogram is a rhombus, though any ......
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
Video: What is a Cone?
A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a solid object; otherwise it is a two-dimensional object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the lateral surface; if the lateral surface is unbounded, it is a conical surface.
In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe.
The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a circular symmetry.
In common usage in elementary geometry, cones are assumed to be right circular, where circular means that the base is a circle and right means that the axis passes through the centre of the base at right angles to its plane. If the cone is right circular the intersection of a plane with the lateral surface is a conic section. In general, however, the base may be any shape and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base). Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly.
A cone with a polygonal base is called a pyramid.
Depending on the context, "cone" may also mean specifically a convex cone or a projective cone.
Volume
Video: Volume of a Cone
The volume of any conic solid is one third of the product of the area of the base and the height
Center of mass
The center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.
Right circular cone
Volume
For a circular cone with radius r and height h, the base is a circle of area and so the formula for volume becomes[6]
Slant height
The slant height of a right circular cone is the distance from any point on the circle of its base to the apex via a line segment along the surface of the cone. It is given by , where is the radius of the base and is the height. This can be proved by the ....
A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.
Video: Geometry: Introduction to Circles - radius, diameter, circumference and area of a circle
A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.
Terminology
Annulus: the ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of the circle.
Centre: the point equidistant from the points on the circle.
Chord: a line segment whose endpoints lie on the circle.
Circumference: the length of one circuit along the circle, or the distance around the circle.
Diameter: a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a line segment, which is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord, and it is twice the radius.
Disc: the region of the plane bounded by a circle.
Lens: the intersection of two discs.
Passant: a coplanar straight line that does not touch the circle.
Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
Sector: a region bounded by two radii and an arc lying between the radii.
Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord's endpoints.
Secant: an extended chord, a coplanar straight line cutting the circle at two points.
Semicircle: an arc that extends from one of a diameter's endpoints to the other. In non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
Tangent: a coplanar straight line that touches the circle at a single point.
Chord, secant, tangent, radius, and diameter
Arc, sector, and segment
Analytic results
Length of circumference
Video: Circumference of a Circle
The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:
The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.
For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.
Video: Median of a Triangle
Finding the median in sets of data with an odd and even number of values
The median is a commonly used measure of the properties of a data set in statistics and probability theory. The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets which vary greatly, a mean may be skewed by a small number of extremely high or low values. Median income, for example, may be a better way to suggest what a "typical" income is.
Because of this, the median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large or small result.
Finite set of numbers
The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest.
If there is an odd number of numbers, the middle one is picked. For example, ......
In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional.
Video: Similar Triangles
It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several statements each of which is necessary and sufficient for two triangles to be similar:
The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:
If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar.
All the corresponding sides have lengths in the same ratio:
AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.
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
of the triangle. If the triangle has equal sides of length 
and base of length 
, the general triangle formulas for the lengths of these segments all simplify to
(half the product of the diagonals)
of any conic solid is one third of the product of the area of the base 
and the height 
and so the formula for volume becomes[6]
, where 
is the radius of the base and 
