15 18 27 triangle

Obtuse scalene triangle.

Sides: a = 15   b = 18   c = 27

Area: T = 127.2799220614
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 31.58663380965° = 31°35'11″ = 0.55112855984 rad
Angle ∠ B = β = 38.9422441269° = 38°56'33″ = 0.68796738189 rad
Angle ∠ C = γ = 109.4711220634° = 109°28'16″ = 1.91106332362 rad

Height: ha = 16.97105627485
Height: hb = 14.14221356237
Height: hc = 9.42880904158

Median: ma = 21.68552484422
Median: mb = 19.98997487421
Median: mc = 9.60546863561

Inradius: r = 4.24326406871
Circumradius: R = 14.3198912319

Vertex coordinates: A[27; 0] B[0; 0] C[11.66766666667; 9.42880904158]
Centroid: CG[12.88988888889; 3.14326968053]
Coordinates of the circumscribed circle: U[13.5; -4.7732970773]
Coordinates of the inscribed circle: I[12; 4.24326406871]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.4143661903° = 148°24'49″ = 0.55112855984 rad
∠ B' = β' = 141.0587558731° = 141°3'27″ = 0.68796738189 rad
∠ C' = γ' = 70.52987793655° = 70°31'44″ = 1.91106332362 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 = 15 ; ; b = 18 ; ; c = 27 ; ;

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

p = a+b+c = 15+18+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 * (30-15)(30-18)(30-27) } ; ; T = sqrt{ 16200 } = 127.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 127.28 }{ 15 } = 16.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 127.28 }{ 18 } = 14.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 127.28 }{ 27 } = 9.43 ; ;

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{ 15**2-18**2-27**2 }{ 2 * 18 * 27 } ) = 31° 35'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-15**2-27**2 }{ 2 * 15 * 27 } ) = 38° 56'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-15**2-18**2 }{ 2 * 18 * 15 } ) = 109° 28'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 127.28 }{ 30 } = 4.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 31° 35'11" } = 14.32 ; ;




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