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   Triangle Formulas   

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also see Defining Geometric Figures

  Triangle
 

 

 

A polygon (plane figure) with
3 angles and 3 sides.

Sides: a, b, c
Opposite angles: A, B, C

Altitudes: ha , hb , hc
Medians: ma , mb , mc
Angle bisectors: ta , tb , tc

Perimeter: P
Semiperimeter: s
Area: K

Radius of circumscribed circle: R
Radius of inscribed circle: r

To read about triangles, visit The Geometry Center.

  Equilateral Triangle

 

   
A triangle with all three sides of equal length.

a = b = c.
A = B = C = Pi/3 radians = 60o

   

 
P = 3a
s = 3a/2
K = a2sqrt(3)/4

ha = ma = ta = a sqrt(3)/2

R = a sqrt(3)/3
r = a sqrt(3)/6

JavaSketchpad exploration: Equilateral triangle



  Isosceles Triangle

   
A triangle with two sides of equal length.

a = c
A = C
 

   

B + 2A = Pi radians = 180o
P = 2a + b
s = a + b/2
K = b sqrt(4a2-b2)/4 = a2 sin(B)/2 = ab sin(A)/2

ha = b sqrt(4a2-b2)/(2a) = a sin(B) = b sin(A)
ma = sqrt(a2+2b2)/2
ta = b sqrt(a[2a+b])/(a+b) = b sin(A)/sin(3A/2)
hb = mb = tb = sqrt(4a2-b2)/2 = a cos(B/2)

R = a2b/4K = a/[2 sin(A)] = b/[2 sin(B)]
r = K/s = b sqrt[(2a-b)/(2a+b)]/2

   

JavaSketchpad exploration: Isosceles triangle



  Right Triangle  
 
A triangle with one right angle.

C = A + B = Pi/2 radians = 90o

c2 = a2 + b2
     (Pythagorean Theorem)
 



 

    P = a + b + c
s = (a+b+c)/2
K = ab/2

ha = b
hb = a
hc = ab/c
ma = sqrt(4b2+a2)/2
mb = sqrt(4a2+b2)/2
mc = c/2
   

ta = 2bc cos(A/2)/(b+c) =
       sqrt[bc(1-a2/[b+c]2)]

tb = 2ac cos(B/2)/(a+c) =
       sqrt[ac(1-b2/[a+c]2)]

tc = ab sqrt(2)/(a+b)

R = c/2
r = ab/(a+b+c) = s - c

 
   

JavaSketchpad exploration: Right triangle



  Scalene Triangle

   
 
A triangle with no two sides equal.

(Note that the following formulas work with all triangles, not just scalene triangles.)

P = a + b + c
s = (a+b+c)/2
 
K = aha/2 = ab sin(C)/2 =
     a2 sin(B) sin(C)/[2 sin(A)] =
     sqrt[s(s-a)(s-b)(s-c)]
     (Heron's or Hero's Formula)
 
ha = c sin(B) = b sin(C) = 2K/a
ma = sqrt(2b2+2c2-a2)/2
ta = 2bc cos(A/2)/(b+c) =
     sqrt[bc(1-a2/[b+c]2)]
 

R = abc/4K =
     a/[2 sin(A)] =
     b/[2 sin(B)] =
     c/[2 sin(C)]
 
r = 2K/P = K/s =
     sqrt[(s-a)(s-b)(s-c)/s] =
     c sin(A/2)sin(B/2)/cos(C/2) =
     ab sin(C)/(2 s) =
     (s-c)tan(C/2)
 

JavaSketchpad exploration: General triangle

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