Lia's Math – Home page

Polygons and Circle – Definitions, Perimeter and Area formulas

Posted in Math Theory by Lia on 04/02/2013

General_2d_figures_1 

 

Below is a table with  perimeter and  area formulas, for the shapes shown on the right hand side.

 

For a table with  definitions, area and volume formulas, for the most common 3D body shapes, click here.

 

 

 

Shape/Description

Visual

representation

Perimeter (p)

Area (A)

Polygon

A closed plane figure that  has three or more straight sides and angles

OR

A polygon with n sides and n angles has the sum of the angles = (n-2)*180˚

Examples of polygons are: Triangle (3 sides); Quadrilateral (4 sides); Pentagon (5 sides); Hexagon (6 sides); Heptagon (7 sides); Octagon (8 sides); Nonagon (9 sides); Decagon (10 sides) and so on.

 

Triangle

A polygon with 3 sides and 3 angles

The sum of the triangle’s angles is equal with 180˚

Scalene

triangle

A triangle with unequal sides and therefore unequal angles.

Obtuse

scalene

 triangle

 A triangle with one of the angles larger than 90˚.    

In this case the angle A is larger than 90˚.

Obtuse triangle_8

 tabular{11}{11}{{p = AB + BC + CA}}

tabular{11}{11}{{A={{h*b}/2}}}

 

Acute

scalene

triangle

A triangle that has no angle larger than 90˚.

Acute triangle_1

 Isosceles triangle

 A triangle that has two sides and therefore two angles equal.

In this case AB = AC and angle B is equal with angle C.

Isosceles triangle_7

Equilateral triangle

A triangle that has all sites and angles equal.

In this case AB = BC =CA and angle A = angle B = angle C.

Equilateral triangle_5

 Right triangle

A triangle that has one of its angle 90˚ therefore the two adjacent sites are perpendicular on each other.

In this case angle A = 90˚ and therefore AC is perpendicular on AB and reverse (AB is perpendicular on AC).

Right triangle_14

Quadrilateral

A polygon with 4 sides and 4 angles.

The sum of the quadrilateral’s angles is equal with 360˚

Quadrilateral

(general shape)

Quadrilateral general_11

tabular{11}{10}{{p = AB + BC +}}

tabular{11}{01}{{CD + DA}}

tabular{11}{10}{{A = Area1 +}}

tabular{11}{01}{{{}{Area2}{}}}

Parallelogram

 A quadrilateral that has the  opposite sides equal and the opposite angles equal.

In this case AB = CD; AC = BD; angle A = angle D and angle B = angle C.

parallelogram_9

tabular{11}{11}{{A = h*b }}

Rhombus

 A parallelogram that has all sides equal.

rhombus_13

tabular{11}{11}{{A = {{d1*d2}/2}}}

Rectangle

 A parallelogram that has all angles equal with 90°.

rectangle_12

tabular{11}{11}{{A = h*b}}

Square

 A rectangle that has all sides equal.

square_15

tabular{11}{11}{{A = a^2 }}

Trapezoid

A quadrilateral with two opposite parallel sides, bases and two lateral sides, legs,  that are not parallel.

 

Scalene trapezoid

 A trapezoid that has no equal  sides or angles.

 trapezoid_16

tabular{11}{11}{{A = {{h({b_1} + {b_2})}/2}}}

Right trapezoid

A trapezoid that has  two adjacent angle equal with 90°.

In this case angle A = angle C = 90°.

 trapezoid_right_18

Isosceles trapezoid

 A trapezoid that has equal legs  and equal adjacent angle.

 In this case AC = BD and angle A= angle B and angle C = angle D.

 trapezoid_isosceles_17

Polygon (general shape)

A polygon with n sides and n angles

has the sum of the angles = (n-2)*180˚

in our example n= 7 therefore the sum of the angles = (7-2)*180^{circ} = 900^{circ}

OR:

There are 5 triangles inside this heptagon (a polygon with 7 sides).  One triangle has the sum of its angle 180^{circ} therefore the polygon will have 5 times the sum of the trianle’s angles (= 5*180^{circ} = 900^{circ}

Polygon (general)_10

tabular{11}{11}{{{p} = {{sum}{}{of}{} {sides}}}}

in our example:

tabular{11}{10}{{p = AB + BC +}}

tabular{11}{00}{{CD + DE +}}

tabular{11}{01}{{EF + FG + GA}}

tabular{11}{10}{{A={sum}{}{of}}}

tabular{11}{01}{{{triangles}{prime}{} {areas}}} (triangles that form the polygon)

In our example:

tabular{11}{10}{{A = Area1 + }}

tabular{11}{00}{{Area2 + Area3 +}}

tabular{11}{01}{{Area4 + Area5}}

Circle

A closed plane curve which points are equidistant from a fixed point, the center.

OR

A set of points equal distanced from a point, the center.

Circle

Circle_4

tabular{11}{11}{{{p} = 2{pi}{R}}}

tabular{11}{11}{{A = {pi}{R^2}}}

(when area is function of R)

or 

tabular{11}{11}{{A = {pi}{{D^2}/4}}}

(when area is function of D)

Circle sector

A circle sector  is the portion of a circle in between two radii.

circle sector_2

tabular{11}{11}{{{p} = a + 2R}}

where:

{}{}{}{a} = 2{pi}{R}*{{alpha}/{{360}^{circ}}} =

{{{pi}R{alpha}}/{{180}^{circ}}}{}{}{}

if {}{alpha}{} is in degrees

and,

{}{}{}{a} = 2{pi}{R}*{{alpha}/{2{pi}}} =

R{alpha}{}{}{}

if {}{alpha}{} is in radians

 

tabular{11}{11}{{A = {pi}{R^2}*{{alpha}/{{360}^{circ}}}}}

if {}{alpha}{} is in degrees

and,

tabular{11}{11}{{A = {{{R^2}{alpha}}/{2}}}}

if {}{alpha}{} is in radians

Circle segment

A circle segment is the portion of a circle in between a chord and the arc that has same end points as the chord.

circle segment_3

tabular{11}{11}{{{p} = a + c}}

where {}{a}{} has same formula as for Circle sector andc = 2Rsin{{alpha}/2}

Look here for sin{{alpha}/2} formula.

tabular{11}{10}{{A = {pi}{R^2}*{{alpha}/{{360}^{circ}}} -}}

tabular{11}{01}{{{1/4}a{sqrt{4{R^2} -{a^2}}}}}

if {}{alpha}{} is in degrees

and,

tabular{11}{10}{{A = {{{R^2}{alpha}}/{2}} -}}

tabular{11}{01}{{{1/4}a{sqrt{4{R^2} -{a^2}}}}}

if {}{alpha}{} is in radians

 

 

 

 

 

Tagged with: ,