12 17 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 17   c = 27

Area: T = 70.21997150991
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 17.81111815332° = 17°48'40″ = 0.31108637614 rad
Angle ∠ B = β = 25.67991768138° = 25°40'45″ = 0.44881861846 rad
Angle ∠ C = γ = 136.5109641653° = 136°30'35″ = 2.38325427076 rad

Height: ha = 11.76999525165
Height: hb = 8.25987900117
Height: hc = 5.21999788962

Median: ma = 21.74985631709
Median: mb = 19.0855334684
Median: mc = 5.85223499554

Inradius: r = 2.50771326821
Circumradius: R = 19.61554642231

Vertex coordinates: A[27; 0] B[0; 0] C[10.81548148148; 5.21999788962]
Centroid: CG[12.60549382716; 1.73333262987]
Coordinates of the circumscribed circle: U[13.5; -14.23108269854]
Coordinates of the inscribed circle: I[11; 2.50771326821]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.1898818467° = 162°11'20″ = 0.31108637614 rad
∠ B' = β' = 154.3210823186° = 154°19'15″ = 0.44881861846 rad
∠ C' = γ' = 43.4990358347° = 43°29'25″ = 2.38325427076 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 = 17 ; ; c = 27 ; ;

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

p = a+b+c = 12+17+27 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-12)(28-17)(28-27) } ; ; T = sqrt{ 4928 } = 70.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 70.2 }{ 12 } = 11.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 70.2 }{ 17 } = 8.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 70.2 }{ 27 } = 5.2 ; ;

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**2-17**2-27**2 }{ 2 * 17 * 27 } ) = 17° 48'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 25° 40'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-17**2 }{ 2 * 17 * 12 } ) = 136° 30'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 70.2 }{ 28 } = 2.51 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 17° 48'40" } = 19.62 ; ;




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