13 20 27 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 20   c = 27

Area: T = 123.6933168769
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 27.26660444507° = 27°15'58″ = 0.47658822497 rad
Angle ∠ B = β = 44.81437424099° = 44°48'49″ = 0.78221473552 rad
Angle ∠ C = γ = 107.9220213139° = 107°55'13″ = 1.88435630487 rad

Height: ha = 19.03297182721
Height: hb = 12.36993168769
Height: hc = 9.16224569458

Median: ma = 22.85327897641
Median: mb = 18.68215416923
Median: mc = 10.11218742081

Inradius: r = 4.12331056256
Circumradius: R = 14.18883340646

Vertex coordinates: A[27; 0] B[0; 0] C[9.22222222222; 9.16224569458]
Centroid: CG[12.07440740741; 3.05441523153]
Coordinates of the circumscribed circle: U[13.5; -4.36656412507]
Coordinates of the inscribed circle: I[10; 4.12331056256]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.7343955549° = 152°44'2″ = 0.47658822497 rad
∠ B' = β' = 135.186625759° = 135°11'11″ = 0.78221473552 rad
∠ C' = γ' = 72.08797868606° = 72°4'47″ = 1.88435630487 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 = 13 ; ; b = 20 ; ; c = 27 ; ;

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

p = a+b+c = 13+20+27 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-13)(30-20)(30-27) } ; ; T = sqrt{ 15300 } = 123.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 123.69 }{ 13 } = 19.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 123.69 }{ 20 } = 12.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 123.69 }{ 27 } = 9.16 ; ;

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{ 13**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 27° 15'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-13**2-27**2 }{ 2 * 13 * 27 } ) = 44° 48'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-13**2-20**2 }{ 2 * 20 * 13 } ) = 107° 55'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 123.69 }{ 30 } = 4.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 27° 15'58" } = 14.19 ; ;




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