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| Quadrilaterals,
Polygons - Regular Polygons, Circle |
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Quadrilaterals |
Square
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Rhombus
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Rectangle |
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Parallelogram |
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Kite
or deltoid
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Trapezium
or trapezoid
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Isosceles
trapezoid
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Cyclic quadrilateral |
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Tangential quadrilateral
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Polygons
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Regular
polygons |
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Circle
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Central angle, inscribed angle
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Construction of a tangent
from a point
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Circumference, length of an arc
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Circle and circular sector
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Circular segment
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Annulus and annulus segment |
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| Quadrilaterals |
| Square
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| A
square is a plane figure with four sides of equal length and
four right angles. |
| The
two diagonals, which are of equal length, bisect each other
perpendicularly. |
| The
square has four axes of symmetry. |
|
 |
 |
| perimeter
P
= 4a |
| area
A
= a ·
a = a2 |
A
= a2 |
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| Rhombus
or rhomb
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| A
rhombus is an oblique-angled parallelogram whose four sides are equal. |
| Rhombus
diagonals bisect each other perpendicularly. |
| The
rhombus is symmetrical about both of its diagonals. |
|
 |
 |
| perimeter
P
= 4a |
| area |
A
= a ·
h and |
 |
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| Rectangle |
| A
rectangle is a plane figure with four straight sides, two parallel pairs
of equal length forming four right |
| angles. |
| The two diagonals of
a rectangle have equal lengths. |
| The
rectangle has two axes of symmetry. |
|
 |
| diagonal |
 |
| perimeter
P
= 2(a + b) |
| area |
|
A
= a ·
b |
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| Parallelogram |
| A
parallelogram
is a quadrilateral (plane figure with four sides) with opposite
sides parallel and of equal |
| length. |
| The
opposite angles of a parallelogram are also equal. |
| In
the case in which the angles are all right angles the
parallelogram is a rectangle, and when all the sides |
| are equal
it is a rhombus. |
|
 |
|
 |
| perimeter
P
= 2(a + b) |
| area |
|
A
= a ·
h |
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| Kite
or deltoid
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| A
kite or deltoid is a quadrilateral with two pairs of equal
adjacent sides. |
| Two
of the angles in a kite are opposite and equal. |
| The kite's diagonals cross perpendicularly, and one of them (the shorter one) is being bisected by the other. |
|
 |
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| perimeter
P
= 2(a + b) |
| area |
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| Trapezium
or trapezoid
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| A
trapezium or trapezoid is a quadrilateral with two parallel
sides of unequal length. |
| The trapezium's median is a straight line joining the midpoints of
nonparallel sides and parallel to the two |
| parallel sides. |
|
 |
| a
+ d = 180°, b + g = 180°
=> a + b + g + d =
360° |
| median
m = 1/2(a +
c),
area |
A
= m ·
h |
| perimeter
P
= a + b + c + d |
| A
= c·h + 1/2(a -
c)·h =
1/2(a +
c)·h |
A
=1/2(a
+ c)·h |
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| Isosceles
trapezoid
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| A trapezoid with the two nonparallel sides of equal length and the angles opposite those sides equal, is called |
| an isosceles trapezoid. |
 |
| a
+ b = 180°, |
|
| median
m = 1/2(a +
c),
area |
A
= m ·
h |
| perimeter
P
= a + 2b
+ c |
| A
= c·h + 1/2(a -
c)·h =
1/2(a +
c)·h |
A
=1/2(a
+ c)·h |
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| Cyclic quadrilateral |
| A
quadrilateral inscribed in a circle, so that all its vertices
lie on the circumference is called a cyclic |
|
quadrilateral. |
| The
opposite angles of a cyclic quadrilateral are supplementary. |
|
 |
| a
+ g = b + d = 180°, |
| d1
· d2
= a · c + b
· d |
| perimeter
P
= a + b
+ c
+ d,
s
= 1/2P |
| area |
 |
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| Tangential quadrilateral
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| A
quadrilateral whose sides all lie tangent to the circle
inscribed within the quadrilateral is called a tangential |
|
quadrilateral. |
|
 |
| s
= a +
c = b + d, |
| perimeter
P
= a + b
+ c
+ d, s
= 1/2P |
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| area |
A
= r · s |
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| Polygons
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| A
polygon is closed plane figure
bounded by a number of straight line segments with the same
number of |
| vertices. |
| The
sum of interior angles is (n
-
2) · 180°,
where n
is the number of sides of a polygon. |
| The
sum of exterior angles of a polygon is 360°. |
|
 |
| dn
- number
of diagonals |
| b'
- the
exterior angle |
| dn
= 1/2 · n · (n
-
3) |
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| Sn=
(n
-
2)
· 180° |
- the
sum of interior angles |
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| Regular
polygons |
| In
a regular polygon all sides are equal and all its angles are
equal. |
| The
exterior angle of a regular polygon is b'
= 360°/n, where n
is the number of sides of a regular polygon. . |
|
 |
| a
=
b'
= 360°/n
|
b'
- the
exterior angle |
| b
=
180°
-
b'
= (n
-
2)
· 180°/n, |
P = n · a |
| dn
= 1/2 · n · (n
-
3) |
- number
of diagonals |
| A
= 1/2 · n · a · r
= 1/2P
· r |
-
area of a regular polygon |
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Circle
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Central angle, inscribed angle
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Construction of a tangent
from a point |
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Circumference, length of an arc
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Circle and circular sector
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Circular segment
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Annulus and annulus segment |
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| Circle
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| A circle is a set of points that are at the fixed distance (called the
radius r) from a fixed point called the |
| center
O. |
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| Central angle, inscribed angle
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| A
central angle is
double the inscribed angle (formed when two secant lines intersect
on the circle) subtended |
| by the same arc. |
| Proof:
Angles
b1
and b2 are
external angles of the isosceles triangle's AOC and
BOC, hence |
| b1
= 2a1,
b2
= 2a2,
a
= a1+
a2
=>
b = b1+
b2
= 2(a1+
a2)
= 2a |
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| Inscribed angles
subtended by the same arc are equal. An angle inscribed in a semicircle is a right angle. |
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| Construction of a tangent
from a point P
to a circle c. |
 |
The
midpoint of the line segment OP
is the circumcenter of the quadrilateral PD1OD2.
The lines PD1
and PD2
are tangents from P
to the given circle c. |
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| Circumference, the length of a circle - the perimeter:
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| Length of an arc:
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| Circle and circular sector
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| By substituting
P = 2pr
and R
= r
in the formula for the area of a regular polygon, obtained is the formula
for |
|
the area of a circle, that is : |
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| Circular segment
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| The portion of a circle bounded by an arc and a chord is called a segment. |
| Symbols
used in the formulas: c
-chord, r
-radius, h
-height of a segment, A
-area of a segment. |
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| Annulus and annulus segment |
| Annulus
or ring is the region enclosed between two concentric circles. |
| Annulus |
Annulus
segment |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
