All geometry formulas for any triangles

Side of a triangle

side arbitrary triangle

 

a , b , c blue    -  sides

alpha red1 , beta green , Y blue    -  angles

 

 

Calculate the side of a triangle if given two other sides and the angle between them (Cosine Rule) ( a  ) :

side arbitrary triangle F cos

 

Calculate the side of a triangle if given side and any two angles (Sine Rule) ( a  ) :

side arbitrary triangle F sin1

side arbitrary triangle F sin2

side arbitrary triangle F sin3

 

 

Height of a triangle 

1.  The height of a triangle if you know all sides  

height arbitrary triangle 2

a , b , c    -  sides

s  - semiperimeter    

s F

h  - height  measured at right angle to the base

 

Calculate the height of a triangle if given sides  ( h  ) :

height arbitrary triangle F11

 

 

2.  The height of a triangle if you know side and angle or area and base  

height arbitrary triangle

b , c    - lateral sides

a black  - base

alpha red , beta blue    -  angles at the base

A18 black  - area of a triangle

h  - height  measured at right angle to the base

 

Calculate the height of a triangle if given side and angle at the base  ( h  ) :

height arbitrary triangle F21   height arbitrary triangle F22

 

Calculate the height of a triangle if given area and base  ( h  ) :

height arbitrary triangle F3

 

 

3.  The height of a triangle if you know sides and radius of the circumcircle  

height arbitrary triangle R

b , c    - lateral sides

R14 red  - radius of the circumcircle of a triangle

O black  - circumcenter    

h  - height  measured at right angle to the base

 

Calculate the height of a triangle if given two lateral sides and radius of the circumcircle  ( h  ) :

height arbitrary triangle F4     

 

 

Bisector of a triangle

Bisector arbitrary triangle

a , b , c blue    -  sides

c1c2 blue  - line segments  obtained by dividing the bisector

Y red  - angle ABC  divided by a bisector in half

s orange  - semiperimeter    

s (abc)2 orange2

L  - bisector  segment |OB|, dividing the angle ABC in half

 

Calculate the length of a bisector of a triangle if given two sides and angle  ( L  ) :

Bisector arbitrary triangle F1

 

Calculate the length of a bisector of a triangle if given all sides ( L  ) :

Bisector arbitrary triangle F3

Bisector arbitrary triangle F2

Bisector arbitrary triangle F4

 

 

Median of a triangle

Median arbitrary triangle

 

a , b , c blue    -  sides

Y red  - angle   ABC

M15 green  - median  segment |OB|, dividing the side   in half

 

 

Calculate the median of a triangle if given two sides and angle  ( M  ) :

Median arbitrary triangle F1

 

Calculate the median of a triangle if given all sides ( M  ) :

Median arbitrary triangle F2

 

 

Sides of an isosceles triangle

sides isosceles triangle

a  - equal sides

b blue  -  base

alpha green  - angles at the base

beta feolet  - angle formed by the equal sides

 

Calculate the length of equal sides if given side (base) and angle  ( a  ) :

sides isosceles triangle F a1

sides isosceles triangle F a2

 

Calculate the length of a side (base) if given equal sides and angle ( b  ) :

sides isosceles triangle F b1

sides isosceles triangle F b2

 

 

Height Bisector and Median of an isosceles triangle

height isosceles triangle

a black  - equal sides

b black  -  base

alpha red  - angles

beta blue  - angle formed by the equal sides

L   - height  =  bisector  =  median

 

Find the length of height = bisector = median if given lateral side and angle at the base ( L  ) :

height isosceles triangle F1

 

Find the length of height = bisector = median if given side (base) and angle at the base ( L  ) :

height isosceles triangle F2

 

Find the length of height = bisector = median if given equal sides and angle formed by the equal sides ( L  ) :

height isosceles triangle F3

 

Find the length of height = bisector = median if given all side ( L  ) :

height isosceles triangle F4

 

 

Sides (legs and hypotenuse) of a right triangle

sides right triangle

a  ,  b -  legs

c blue  -  hypotenuse

alpha red  ,   beta green - acute angles at the hypotenuse

 

 

Calculate the length of a leg if given other sides and angles  ( a b  ) :

sides right triangle F11   sides right triangle F21

sides right triangle F12   sides right triangle F22

sides right triangle F13   sides right triangle F23

 

Calculate the length of a hypotenuse if given legs and angles at the hypotenuse  ( c  ) :

sides right triangle F31   sides right triangle F32

sides right triangle F41   sides right triangle F42

 

Calculate the length of sides of a right triangle using Pythagorean theorem  ( c  a  b  ) :

sides right triangle F5

sides right triangle F6

sides right triangle F7

 

 

Height of a right triangle 

1.  The height of a right triangle if you know sides and angles  

height right triangle1

a black  ,  b black -  legs of a right triangle

c black  -  hypotenuse

alpha red  ,   beta blue - acute angles at the hypotenuse

h  - height  from the vertex of the right angle

 

Find the length of height if given all sides  ( h  ) :

height right triangle F1

 

Find the length of height if given hypotenuse and angles at the hypotenuse  ( h  ) :

height right triangle F21

height right triangle F22

 

Find the length of height if given legs and angles at the hypotenuse  ( h  ) :

height right triangle F31   height right triangle F32

 

 

2.  The height of a triangle if you know segments of the hypotenuse obtained by dividing the height  

height right triangle3

 

c black  -  hypotenuse

c1c2 blue  - segments  obtained by dividing the height

h  - height  from the vertex of the right angle

 

Find the length of height if given segments of the hypotenuse obtained by dividing the height  ( h  ) :

height right triangle F5

 

 

Bisector of a right triangle 

1.  The bisector of a right triangle, from the vertex of the right angle if you know sides and angle  

Bisector right triangle

a black  ,  b black -  legs

c blue  -  hypotenuse

alpha red  - acute angle at the hypotenuse

L  - bisector  from the vertex of the right angle

 

Calculate the length of a bisector if given legs  ( L  ) :

Bisector right triangle F1

 

Calculate the length of bisector if given hypotenuse and angle at the hypotenuse  ( L  ) :

Bisector right triangle F2

 

 

 

2.  The bisector of a right triangle, from the vertex of the acute angle if you know sides and angles  

Bisector right triangle1

a black  ,  b black  -  legs

c blue  -  hypotenuse

alpha red1  ,  beta feolet  -   acute angles at the hypotenuse

L  - bisector  from the vertex of the acute angle

 

Calculate the length of a bisector if given leg and angles at the hypotenuse ( L  ) :

Bisector right triangle1 F1

Bisector right triangle1 F21

Bisector right triangle1 F22

 

Calculate the length of a bisector if given leg and hypotenuse  ( L  ) :

Bisector right triangle1 F3

 

 

Median of a right triangle

Median right triangle

a12 green  ,  b -  legs

c blue  -  hypotenuse

alpha brown  - acute angle at the hypotenuse

R14 red  - radius  of Circumcircle

DD blue  - diameter  of Circumcircle

M14 red  - median  from the vertex of the right angle

 

The median equals the radius of Circumcircle and the half-hypotenuse  ( M  ) :

Median right triangle F1

 

Calculate the length of median if given legs  ( M  ) :

Median right triangle F2

 

Calculate the length of median if given leg and angle at the hypotenuse ( M  ) :

Median right triangle F31   Median right triangle F32

 

 

Height Bisector and Median of an equilateral triangle

height equilateral triangle

 

a  - equal sides

L   - height  =  bisector  =  median

 

 

 

Find the length of height = bisector = median if given side  ( L  ) :

height equilateral triangle F