18 20 27 triangle

Obtuse scalene triangle.

Sides: a = 18   b = 20   c = 27

Area: T = 179.996565967
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 41.80990791939° = 41°48'33″ = 0.73297060892 rad
Angle ∠ B = β = 47.79330302502° = 47°47'35″ = 0.83441457374 rad
Angle ∠ C = γ = 90.39878905558° = 90°23'52″ = 1.57877408271 rad

Height: ha = 209.9995177411
Height: hb = 187.999565967
Height: hc = 13.33330118274

Median: ma = 21.98986334273
Median: mb = 20.65218764281
Median: mc = 13.40770876778

Inradius: r = 5.53883279898
Circumradius: R = 13.55003255326

Vertex coordinates: A[27; 0] B[0; 0] C[12.09325925926; 13.33330118274]
Centroid: CG[13.03108641975; 4.44443372758]
Coordinates of the circumscribed circle: U[13.5; -0.09437522606]
Coordinates of the inscribed circle: I[12.5; 5.53883279898]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.1910920806° = 138°11'27″ = 0.73297060892 rad
∠ B' = β' = 132.207696975° = 132°12'25″ = 0.83441457374 rad
∠ C' = γ' = 89.60221094442° = 89°36'8″ = 1.57877408271 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 = 18 ; ; b = 20 ; ; c = 27 ; ;

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

p = a+b+c = 18+20+27 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-18)(32.5-20)(32.5-27) } ; ; T = sqrt{ 32398.44 } = 180 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 180 }{ 18 } = 20 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 180 }{ 20 } = 18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 180 }{ 27 } = 13.33 ; ;

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{ 18**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 41° 48'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-18**2-27**2 }{ 2 * 18 * 27 } ) = 47° 47'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-18**2-20**2 }{ 2 * 20 * 18 } ) = 90° 23'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 180 }{ 32.5 } = 5.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 41° 48'33" } = 13.5 ; ;




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