12 16 23 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 16   c = 23

Area: T = 90.42108908383
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 29.43438177134° = 29°26'2″ = 0.51437170305 rad
Angle ∠ B = β = 40.93766253727° = 40°56'12″ = 0.71444788974 rad
Angle ∠ C = γ = 109.6329556914° = 109°37'46″ = 1.91333967257 rad

Height: ha = 15.07701484731
Height: hb = 11.30326113548
Height: hc = 7.86326861599

Median: ma = 18.88112075885
Median: mb = 16.50875740192
Median: mc = 8.23110388166

Inradius: r = 3.54659172878
Circumradius: R = 12.21095678307

Vertex coordinates: A[23; 0] B[0; 0] C[9.06552173913; 7.86326861599]
Centroid: CG[10.68884057971; 2.62108953866]
Coordinates of the circumscribed circle: U[11.5; -4.10216516931]
Coordinates of the inscribed circle: I[9.5; 3.54659172878]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.5666182287° = 150°33'58″ = 0.51437170305 rad
∠ B' = β' = 139.0633374627° = 139°3'48″ = 0.71444788974 rad
∠ C' = γ' = 70.37704430861° = 70°22'14″ = 1.91333967257 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 = 23 ; ;

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

p = a+b+c = 12+16+23 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-12)(25.5-16)(25.5-23) } ; ; T = sqrt{ 8175.94 } = 90.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 90.42 }{ 12 } = 15.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 90.42 }{ 16 } = 11.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 90.42 }{ 23 } = 7.86 ; ;

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**2-23**2 }{ 2 * 16 * 23 } ) = 29° 26'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-12**2-23**2 }{ 2 * 12 * 23 } ) = 40° 56'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-12**2-16**2 }{ 2 * 16 * 12 } ) = 109° 37'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 90.42 }{ 25.5 } = 3.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 29° 26'2" } = 12.21 ; ;




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