12 13 21 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 13   c = 21

Area: T = 71.13436769751
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 31.40878008669° = 31°24'28″ = 0.54881695359 rad
Angle ∠ B = β = 34.37112560241° = 34°22'17″ = 0.65998915857 rad
Angle ∠ C = γ = 114.2210943109° = 114°13'15″ = 1.9943531532 rad

Height: ha = 11.85656128292
Height: hb = 10.94436426116
Height: hc = 6.77546359024

Median: ma = 16.40112194669
Median: mb = 15.81992920196
Median: mc = 6.80107352544

Inradius: r = 3.09327685641
Circumradius: R = 11.51435338819

Vertex coordinates: A[21; 0] B[0; 0] C[9.90547619048; 6.77546359024]
Centroid: CG[10.30215873016; 2.25882119675]
Coordinates of the circumscribed circle: U[10.5; -4.72435010798]
Coordinates of the inscribed circle: I[10; 3.09327685641]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.5922199133° = 148°35'32″ = 0.54881695359 rad
∠ B' = β' = 145.6298743976° = 145°37'43″ = 0.65998915857 rad
∠ C' = γ' = 65.7799056891° = 65°46'45″ = 1.9943531532 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 = 13 ; ; c = 21 ; ;

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

p = a+b+c = 12+13+21 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-12)(23-13)(23-21) } ; ; T = sqrt{ 5060 } = 71.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 71.13 }{ 12 } = 11.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 71.13 }{ 13 } = 10.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 71.13 }{ 21 } = 6.77 ; ;

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**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 31° 24'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 34° 22'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 114° 13'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 71.13 }{ 23 } = 3.09 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 31° 24'28" } = 11.51 ; ;




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