12 20 30 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 20   c = 30

Area: T = 80.49222356504
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 15.56435751871° = 15°33'49″ = 0.27216356304 rad
Angle ∠ B = β = 26.56328406278° = 26°33'46″ = 0.46436090276 rad
Angle ∠ C = γ = 137.8743584185° = 137°52'25″ = 2.40663479956 rad

Height: ha = 13.41553726084
Height: hb = 8.0499223565
Height: hc = 5.36661490434

Median: ma = 24.77990233867
Median: mb = 20.54326385842
Median: mc = 6.85656546004

Inradius: r = 2.59765237307
Circumradius: R = 22.36224053358

Vertex coordinates: A[30; 0] B[0; 0] C[10.73333333333; 5.36661490434]
Centroid: CG[13.57877777778; 1.78987163478]
Coordinates of the circumscribed circle: U[15; -16.58554506241]
Coordinates of the inscribed circle: I[11; 2.59765237307]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.4366424813° = 164°26'11″ = 0.27216356304 rad
∠ B' = β' = 153.4377159372° = 153°26'14″ = 0.46436090276 rad
∠ C' = γ' = 42.12664158149° = 42°7'35″ = 2.40663479956 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 12 ; ; b = 20 ; ; c = 30 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 12+20+30 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-12)(31-20)(31-30) } ; ; T = sqrt{ 6479 } = 80.49 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80.49 }{ 12 } = 13.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80.49 }{ 20 } = 8.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80.49 }{ 30 } = 5.37 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 12**2-20**2-30**2 }{ 2 * 20 * 30 } ) = 15° 33'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-12**2-30**2 }{ 2 * 12 * 30 } ) = 26° 33'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-12**2-20**2 }{ 2 * 20 * 12 } ) = 137° 52'25" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80.49 }{ 31 } = 2.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 15° 33'49" } = 22.36 ; ;




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