12 16 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 16   c = 27

Area: T = 49.50769439574
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 13.25499072144° = 13°15' = 0.23112545065 rad
Angle ∠ B = β = 17.79441698701° = 17°47'39″ = 0.31105668519 rad
Angle ∠ C = γ = 148.9565922916° = 148°57'21″ = 2.65997712952 rad

Height: ha = 8.25111573262
Height: hb = 6.18883679947
Height: hc = 3.66771810339

Median: ma = 21.36658606192
Median: mb = 19.33002590656
Median: mc = 4.21330748866

Inradius: r = 1.88002525075
Circumradius: R = 26.178814586

Vertex coordinates: A[27; 0] B[0; 0] C[11.42659259259; 3.66771810339]
Centroid: CG[12.80986419753; 1.2222393678]
Coordinates of the circumscribed circle: U[13.5; -22.42986718436]
Coordinates of the inscribed circle: I[11.5; 1.88002525075]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.7550092786° = 166°45' = 0.23112545065 rad
∠ B' = β' = 162.206583013° = 162°12'21″ = 0.31105668519 rad
∠ C' = γ' = 31.04440770845° = 31°2'39″ = 2.65997712952 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 = 16 ; ; c = 27 ; ;

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

p = a+b+c = 12+16+27 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-12)(27.5-16)(27.5-27) } ; ; T = sqrt{ 2450.94 } = 49.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 49.51 }{ 12 } = 8.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 49.51 }{ 16 } = 6.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 49.51 }{ 27 } = 3.67 ; ;

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**2-16**2-27**2 }{ 2 * 16 * 27 } ) = 13° 15' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 17° 47'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-16**2 }{ 2 * 16 * 12 } ) = 148° 57'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 49.51 }{ 27.5 } = 1.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 13° 15' } = 26.18 ; ;




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