12 23 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 23   c = 27

Area: T = 137.2888018414
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 26.24112314813° = 26°14'28″ = 0.45879958891 rad
Angle ∠ B = β = 57.9366136214° = 57°56'10″ = 1.01111763328 rad
Angle ∠ C = γ = 95.82326323048° = 95°49'21″ = 1.67224204316 rad

Height: ha = 22.88113364023
Height: hb = 11.93880885577
Height: hc = 10.16994828455

Median: ma = 24.35215913238
Median: mb = 17.44327635425
Median: mc = 12.42197423484

Inradius: r = 4.42986457553
Circumradius: R = 13.57700115824

Vertex coordinates: A[27; 0] B[0; 0] C[6.37703703704; 10.16994828455]
Centroid: CG[11.12334567901; 3.39898276152]
Coordinates of the circumscribed circle: U[13.5; -1.37766678417]
Coordinates of the inscribed circle: I[8; 4.42986457553]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.7598768519° = 153°45'32″ = 0.45879958891 rad
∠ B' = β' = 122.0643863786° = 122°3'50″ = 1.01111763328 rad
∠ C' = γ' = 84.17773676952° = 84°10'39″ = 1.67224204316 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 = 23 ; ; c = 27 ; ;

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

p = a+b+c = 12+23+27 = 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)(31-23)(31-27) } ; ; T = sqrt{ 18848 } = 137.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 137.29 }{ 12 } = 22.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 137.29 }{ 23 } = 11.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 137.29 }{ 27 } = 10.17 ; ;

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**2-23**2-27**2 }{ 2 * 23 * 27 } ) = 26° 14'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 57° 56'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-23**2 }{ 2 * 23 * 12 } ) = 95° 49'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 137.29 }{ 31 } = 4.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 26° 14'28" } = 13.57 ; ;




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