This document discusses different types of triangles and their properties. It explains that all triangles have 3 sides and 3 angles that add up to 180 degrees. It also discusses the Triangle Inequality Theorem. Triangles are classified based on equal or unequal side lengths (scalene, isosceles, equilateral) and angle types (acute, right, obtuse). When combining these classifications, there are 7 possible triangle types. Examples and diagrams of each type are provided.

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2.  All triangles have 3 sides and 3 angles which always
add up to 180°.
 The Triangle Inequality Theorem states that:
The longest side of any triangle must be
less than the sum of the other 2 sides.
 Triangles are classified in 2 ways-
 1) By the number of equal sides they have:
• scalene - all 3 sides have different lengths
• isosceles - 2 sides have equal lengths
• equilateral - all 3 sides are equal


3.  2) By the types of angles they have:
• acute triangle - all 3 angles are acute (less than
90°)
• right triangle - has one right angle (a right angle
= 90°)
• obtuse triangle - has one obtuse angle
(an obtuse angle is greater than 90° and less than
180°).
 When these 2 categories are combined, there are 7
possible triangles:
• acute scalene (diagram A)
• right scalene (B) - all right triangles are scalene
(except diagram E).
• obtuse scalene (C)

4. • acute isosceles (diagram D)
• right isosceles (E) also
known as a 45° 45° 90°
triangle.
• obtuse isosceles (F)

5.  • equilateral (G) all sides are equal and each angle
= 60°, making this the only equiangular triangle.
Since all 3 angles are less than 90° all equilateral
triangles are acute triangles.

7.  his article is about four-sided mathematical shapes. For other uses,
see Quadrilateral (disambiguation).
 Quadrilateral
Six different types of quadrilaterals
 Edges and vertices4Schläfli symbol{4} (for square)Areavarious methods;
see belowInternal angle(degrees)90° (for square and rectangle)In Euclidean
plane geometry, a quadrilateral is a polygon with four sides (or edges) and
four vertices or corners. Sometimes, the term quadrangleis used, by
analogy with triangle, and sometimes tetragon for consistency
with pentagon (5-sided), hexagon (6-sided) and so on.
 The origin of the word "quadrilateral" is the two Latin words quadri, a
variant of four, and latus, meaning "side."
 Quadrilaterals are simple (not self-intersecting) or complex (self-
intersecting), also called crossed. Simple quadrilaterals are
either convex or concave.
 The interior angles of a simple (and planar) quadrilateral ABCD add up to
360 degrees of arc, that is
 This is a special case of the n-gon interior angle sum formula (n − 2) × 180°.
In a crossed quadrilateral, the four interior angles on either side of the
crossing add up to 720°.[1]
 All convex quadrilaterals tile the plane by repeated rotation around the
midpoints of their edges.

8.  Euler diagram of some types of quadrilaterals. (UK) denotes British English
and (US) denotes American English.
 A parallelogram is a quadrilateral with two pairs of parallel sides.
Equivalent conditions are that opposite sides are of equal length; that
opposite angles are equal; or that the diagonals bisect each other.
Parallelograms also include the square, rectangle, rhombus and rhomboid.
 Rhombus or rhomb: all four sides are of equal length. An equivalent
condition is that the diagonals perpendicularly bisect each other. An
informal description is "a pushed-over square" (including a square).
 Rhomboid: a parallelogram in which adjacent sides are of unequal lengths
and angles are oblique (not right angles). Informally: "a pushed-over
rectangle with no right angles."[2]
 Rectangle: all four angles are right angles. An equivalent condition is that
the diagonals bisect each other and are equal in length. Informally: "a box
or oblong" (including a square).
 Square (regular quadrilateral): all four sides are of equal length
(equilateral), and all four angles are right angles. An equivalent condition
is that opposite sides are parallel (a square is a parallelogram), that the
diagonals perpendicularly bisect each other, and are of equal length. A
quadrilateral is a square if and only if it is both a rhombus and a rectangle
(four equal sides and four equal angles).
 .

9. Oblong: a term sometimes used to denote a rectangle which has unequal
adjacent sides (i.e. a rectangle that is not a square).[3]
Kite: two pairs of adjacent sides are of equal length. This implies that one
diagonal divides the kite into congruent triangles, and so the angles between
the two pairs of equal sides are equal in measure. It also implies that the
diagonals are perpendicular.
Right kite: a kite with two opposite right angles.
Trapezoid (North American English)
or Trapezium (British English): at least one pair of opposite sides are parallel.
Trapezium (NAm.): no sides are parallel. (In British English this would be
called an irregular quadrilateral, and was once called a trapezoid.)
Isosceles trapezoid (NAm.) or isosceles trapezium (Brit.): one pair of opposite
sides are parallel and the base angles are equal in measure. Alternative
definitions are a quadrilateral with an axis of symmetry bisecting one pair of
opposite sides, or a trapezoid with diagonals of equal length.
Tangential trapezoid: a trapezoid where the four sides are tangents to
an inscribed circle

10.  Tangential quadrilateral: the four sides are tangents to an
inscribed circle. A convex quadrilateral is tangential if and only if
opposite sides have equal sums.
 Cyclic quadrilateral: the four vertices lie on a circumscribed
circle. A convex quadrilateral is cyclic if and only if opposite
angles sum to 180°.
 Bicentric quadrilateral: it is both tangential and cyclic.
 Orthodiagonal quadrilateral: the diagonals cross at right angles.
 Equidiagonal quadrilateral: the diagonals are of equal length.
 Ex-tangential quadrilateral: the four extensions of the sides are
tangent to an excircle.
 An equilic quadrilateral has two opposite equal sides that, when
extended, meet at 60°.
 A Watt quadrilateral is a quadrilateral with a pair of opposite
sides of equal length.[4]

11.  A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter
of a square.[5]
 A geometric chevron (dart or arrowhead) is a concave quadrilateral with bilateral symmetry
like a kite, but one interior angle is reflex.
 A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed
quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. A special case of crossed
quadrilaterals are the antiparallelograms, crossed quadrilaterals in which (like
a parallelogram) each pair of nonadjacent sides has equal length. The diagonals of a crossed
or concave quadrilateral do not intersect inside the shape.
 A non-planar quadrilateral is called a skew quadrilateral. Formulas to compute its dihedral
angles from the edge lengths and the angle between two adjacent edges were derived for
work on the properties of molecules such as cyclobutane that contain a "puckered" ring of four
atoms.[6] See skew polygon for more. Historically the term gauche quadrilateral was also used
to mean a skew quadrilateral.[7] A skew quadrilateral together with its diagonals form a
(possibly non-regular) tetrahedron, and conversely every skew quadrilateral comes from a
tetrahedron where a pair of opposite edges is removed.
 The two diagonals of a convex quadrilateral are the line segments that connect opposite
vertices.
 The two bimedians of a convex quadrilateral are the line segments that connect the midpoints
of opposite sides.[8] They intersect at the "vertex centroid" of the quadrilateral
(see Remarkable points below).
 The four maltitudes of a convex quadrilateral are the perpendiculars to a
side through the midpoint of the opposite side .[9] .

12.  Base: the bottom surface of a solid object.
 Edge: the intersection of two faces on a solid
object. This is a line.
 Face: a flat side of a 3‐dimensional object.
 Prism: a solid object with two congruent and
parallel faces.
 Pyramid: a solid object with a polygon for a base
and triangles for sides.

13.  A circle is a simple shape of Euclidean geometry that is the
set of all points in a plane that are at a given distance from
a given point, the centre. The distance between any of the
points and the centre is called the radius. It can also be
defined as the locus of a point equidistant from a fixed
point.
 A circle is a simple closed curve which 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
the former and the latter is called a disk.
 A circle can be defined as the curve traced out by a point
that moves so that its distance from a given point is
constant.
 A circle may also be defined as a special ellipse in which
the two foci are coincident and the eccentricity is 0.

15.  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.
 Circular sector: a region bounded by two radii and an arc lying between the radii.
 Circular segment: a region, not containing the centre, bounded by a chord and an arc
lying between the chord's endpoints.
 Circumference: the length of one circuit along 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.
 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.
 Secant: an extended chord, a coplanar straight line cutting the circle at two points.
 Semicircle: a region bounded by a diameter and an arc lying between the diameter's
endpoints. It is a special case of a circular segment, namely the largest one.
 Tangent: a coplanar straight line that touches the circle at a single point.


16.  Further information: Circumference
 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:

17.  Area enclosed by a circle = π × area of the shaded
square
 Main article: Area of a disk
 As proved by Archimedes, the area enclosed by a
circleis equal to that of a triangle whose base has the
length of the circle's circumference and whose height
equals the circle's radius,[6] which comes
to π multiplied by the radius squared:
Equivalently, denoting diameter by d

18.  that is, approximately 79 percent of
the circumscribingsquare (whose side is of
length d).
 The circle is the plane curve enclosing the
maximum area for a given arc length. This relates
the circle to a problem in the calculus of
variations, namely
 the isoperimetric inequality

19.  Circle of radius r = 1, centre (a, b) = (1.2, −0.5)
 In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of
all points (x, y) such that
 This equation, also known as Equation of the Circle, follows from the Pythagorean theorem applied to any
point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled
triangle whose other sides are of length x − a and y − b. If the circle is centred at the origin (0, 0), then
the equation simplifies to
 The equation can be written in parametric form using the trigonometric functions sine and cosine as
 where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray
from (a, b) to (x, y) makes with the x-axis. An alternative parametrisation of the circle is:

20.  In this parametrisation, the ratio of t to r can be interpreted
geometrically as the stereographic projection of the circle onto
the line passing through the centre parallel to the x-axis.
 In homogeneous coordinates each conic section with equation of
a circle is of the form
 It can be proven that a conic section is a circle exactly when it
contains (when extended to the complex projective plane) the
points I(1: i: 0) and J(1: −i: 0). These points are called
the circular points at infinity.
 In polar coordinates the equation of a circle is:
 where a is the radius of the circle, is the polar coordinate of a
generic point on the circle, and is the polar coordinate of the
centre of the circle (i.e., r0 is the distance from the origin to the
centre of the circle, and φ is the anticlockwise angle from the
positive x-axis to the line connecting the origin to the centre of
the circle). For a circle centred at the origin, i.e. r0 = 0, this
reduces to simply r = a. When r0 = a, or when the origin lies

21.  on the circle, the equation becomes
 In the general case, the equation can be solved for r, giving
 the solution with a minus sign in front of the square root giving the same curve.
 In the complex plane, a circle with a centre at c and radius (r) has the equation . In
parametric form this can be written .
 The slightly generalised equation for real p, q and
complex g is sometimes called a generalised circle. This becomes the above
equation for a circle
with since . Not all
generalised circles are actually circles: a generalised circle is either a (true) circle
or a line.
 Main article: Tangent lines to circles
 The tangent line through a point P on the circle is perpendicular to the diameter
passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then
the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the
form (x1 − a)x + (y1 – b)y = c. Evaluating at (x1, y1) determines the value of c and
the result is that the equation of the tangent is

22.  Or

If y1 ≠ b then the slope of this line is
 This can also be found using implicit
differentiation.
 When the centre of the circle is at the origin
then the equation of the tangent line
becomes
 and its slope is

23.  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.
 More precisely, it is the solid figure bounded by a base in a
plane and by a surface (called the lateral surface) formed
by the locus of all straight line segments joining the apex to
the perimeter of the base. The term "cone" sometimes
refers just to the surface of this solid figure, or just to the
lateral surface.
 The axis of a cone is the straight line (if any), passing
through the apex, about which the base has a rotational
symmetry.
 In common usage in elementary geometry, cones are
assumed to be right circular, where circular means that the
base is a circle and rightmeans that the axis passes through
the centre of the base at right angles to its plane.
Contrasted with right cones are oblique cones, in which the
axis does not pass perpendicularly through the centre of the
base.[1] 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).

24.  A cylinder (from Greek κύλινδρος – kulindros, "roller,
tumbler"[1]) is one of the most basic curvilinear geometric
shapes, the surface formed by the points at a fixed
distance from a given line segment, the axis of the
cylinder. The solid enclosed by this surface and by two
planes perpendicular to the axis is also called a cylinder.
The surface area and the volume of a cylinder have been
known since deep antiquity.
 In differential geometry, a cylinder is defined more
broadly as any ruled surface spanned by a one-parameter
family of parallel lines. A cylinder whosecross section is
an ellipse, parabola, or hyperbola is called an elliptic
cylinder, parabolic cylinder, or hyperbolic
cylinder respectively.
 The open cylinder is topologically equivalent to both the
open annulus and the punctured plane.

25.  A sphere (from Greek σφαῖρα — sphaira, "globe, ball"[1]) is
a perfectly round geometrical and circular object in three-
dimensional space that resembles the shape of a
completely round ball. Like a circle, which, in geometrical
contexts, is in two dimensions, a sphere is defined
mathematically as the set of points that are all the same
distance r from a given point in three-dimensional space.
This distance r is the radius of the sphere, and the given
point is the center of the sphere. The maximum straight
distance through the sphere passes through the center and
is thus twice the radius; it is the diameter.
 In mathematics, a distinction is made between the sphere
(a two-dimensional closed surface embedded in three-
dimensional Euclidean space) and theball (a three-
dimensional shape that includes the interior of a sphere).