Geometry formulas

Plane Geometry (triangle, parallelogram, square, trapezoid), Solid Geometry (surface formulas bodies)

Plane Geometry

Triangle

The perimeter = the sum of all sides, i.e.:
P=AB+BC+CA

Surface = (height x base)/2, i.e.:
Striangle = (b x h)/2.
In our case, b=BC, and h=AD. So though,
SABC=(BCxAD)/2

 

Parallelogram


The perimeter = the sum of all sides, i.e.:

P=AB + BC + CD + DA.

Since opposite sides of the parallelogram are congruent (equal), the perimeter can be calculated as

P=2(AB + BC).

Parallelogram's Surface = base x height, i.e. Sparallelogram=b x h, and on our case,
SABCD=DC x AM, because
DC = b (base) and AM = h (height).

 

Rectangle

The rectangle has length( L=AB) and width(l=BC).

The perimeter = the sum of all sides, i.e.:

P=AB+BC+CD+DA

or

P=2(L+l)

Rectangle's Surface = length x width
Srectangle = L x l. In our particular case SABCD=AB x BC.

 

Square

The square is a rectangle that has all sides equal (congruent), equal to the width or length.

The perimeter = the sum of all sides, i.e.:

P=AB+BC+CD+DA or P=4 L, where L is sides of the square (AB=BC=CD=DA=L).
Square's Surface L2.
In our case,

SABCD=AB2.

 

Trapezium

The perimeter = the sum of all sides, i.e.:

P=AB + BC + CD + DA.

Trapezium Area = (Big Base + Small Base) x Height / 2, i.e Atrapezium=(B + b) x h/2, and for our figure we have
AABCD = (DC + AB) x AM/2, because
DC = B (Big Base)
AB = b  (Small Base), and
AM = h (Height).

 

The Circle

We have OA - the Radius (not. r)

The Circle's Length (the circumference of the circle):
   
The Circle's Area (Surface):
   

Solid Geometry

Solids -Polyhedra

The Pyramid

We discuss about regular solids, and so is regular pyramid.
We have: AB - the base edge(not. m)
VA - the laterla edge(not. l)
VO - the height of the Pyramid (not. h)
VM - 
apotema side or pyramid apotema (not. ap)
OM - apotema base (not. ab).
The Lateral Area (of the Pyramid) =
sum of the areas of the side faces     

Alat = (Pb x ap) / 2.


The Base Area
     Ab = (Pb x ab) / 2, where Pb is base's perimeter
The Total Area = Base Area + Lateral Area

The Volume
     Vpir=(Ab x h) / 3.
Tetrahedron can be considered a pyramid whose base a triangle, area and volume are computed analog.

 

Cuboid, parallelepiped, prism


We have: AB - length(not. L)
BC - width(not. l)
AE - height or lateral edge (not. h)
     Lateral Surface (Area) = The sum of all lateral sides
     Alat = Pb x h, where Pb is the base perimeter,
or
     Alat = 2(L + l) x h


The Base Area
     Ab = L x l.
Total Area (Surface) = Base Area + Lateral Area

The Volume
     Vcuboid = Ab x h

or

    Vcuboid = L x l x h.

Cuboid is a particular case of light and the cube is a cuboid particular case, in that it is a cuboid with all sides congruent. Therefore we do not remember anything about them here.

 

The Truncated Pyramid


We have: AB - The Edge of Big Base
A'B' - The Edge of Small (Top) Base
OO' - Height (not. h)
AA' - The Lateral Edge
OM - The Apotema of Big Base (not. aB)
O'M' - The Apotema of Small Base (not. ab)
MM' - The Apotema of the Truncated Pyramid (not. at)
     The Lateral Area (Surface) = The sum of all lateral sides Areas
     Alat = (PB + Pb) at / 2, where Pb is the perimeter of small base, and PB is the perimeter of big base.

     Ab=Pb x ab.
     AB=PB x aB.
Total Area = Big Base Area + Small Base Area + Lateral Area

The Volume
     Vtruncated pyramid =

 

Solids - Rounded Solids

The cilinder


We have:
AA' - generator edge (not. g)
OO' - the height of the cilinder (not. h; at our particular case,
at right circular cylinder, we have g=h)
AO - base radius (not. r)
Base Area = the area of the base circle, i.e.:
    
Lateral Area:
    
Total Area:
    
The Volume of the Cilinder:
    

 

The Cone


We have:
VA - the generator (not. g)
VO - Cone's Height (not. h)
AO - Base's radius (not. r)
Base Area = The surface of base circle, i.e.:
    
Lateral Area:
    
Total Area:
    
The Volume:
    

Frustoconic


We have:
A'A - generator (not. G)
OO' - frustoconic's Height (not. I)
AO - Big Base Radius (not. R)
A'O' - Small Base Radius(not. r)
Lateral Area:

Total Area:

The Volume:

 

 

The Sphere


We have:
OA - Radius (not. r)
Area of the Sphere:
    
The Volume:
    

 

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Keywords: formulas geometry, geometric body volume formula, trapezoid, square, rectangle, triangle, sphere

 

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