12 16 21 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 16   c = 21

Area: T = 95.45112310031
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 34.62221618397° = 34°37'20″ = 0.60442707183 rad
Angle ∠ B = β = 49.24986367043° = 49°14'55″ = 0.86595508626 rad
Angle ∠ C = γ = 96.12992014559° = 96°7'45″ = 1.67877710727 rad

Height: ha = 15.90985385005
Height: hb = 11.93114038754
Height: hc = 9.09105934289

Median: ma = 17.67876695297
Median: mb = 15.11662164578
Median: mc = 9.47436476607

Inradius: r = 3.89659686124
Circumradius: R = 10.56603666858

Vertex coordinates: A[21; 0] B[0; 0] C[7.83333333333; 9.09105934289]
Centroid: CG[9.61111111111; 3.03301978096]
Coordinates of the circumscribed circle: U[10.5; -1.12875391513]
Coordinates of the inscribed circle: I[8.5; 3.89659686124]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.378783816° = 145°22'40″ = 0.60442707183 rad
∠ B' = β' = 130.7511363296° = 130°45'5″ = 0.86595508626 rad
∠ C' = γ' = 83.87107985441° = 83°52'15″ = 1.67877710727 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 = 21 ; ;

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

p = a+b+c = 12+16+21 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-12)(24.5-16)(24.5-21) } ; ; T = sqrt{ 9110.94 } = 95.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 95.45 }{ 12 } = 15.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 95.45 }{ 16 } = 11.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 95.45 }{ 21 } = 9.09 ; ;

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**2-21**2 }{ 2 * 16 * 21 } ) = 34° 37'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 49° 14'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-12**2-16**2 }{ 2 * 16 * 12 } ) = 96° 7'45" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 95.45 }{ 24.5 } = 3.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 34° 37'20" } = 10.56 ; ;




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