9 16 21 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 16   c = 21

Area: T = 67.14216413264
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 23.55664643091° = 23°33'23″ = 0.41111378623 rad
Angle ∠ B = β = 45.27550868451° = 45°16'30″ = 0.79901993346 rad
Angle ∠ C = γ = 111.1688448846° = 111°10'6″ = 1.94402554567 rad

Height: ha = 14.92203647392
Height: hb = 8.39327051658
Height: hc = 6.39444420311

Median: ma = 18.1187670932
Median: mb = 14.03656688476
Median: mc = 7.63221687612

Inradius: r = 2.91992017968
Circumradius: R = 11.26597783591

Vertex coordinates: A[21; 0] B[0; 0] C[6.33333333333; 6.39444420311]
Centroid: CG[9.11111111111; 2.1311480677]
Coordinates of the circumscribed circle: U[10.5; -4.06660310741]
Coordinates of the inscribed circle: I[7; 2.91992017968]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.4443535691° = 156°26'37″ = 0.41111378623 rad
∠ B' = β' = 134.7254913155° = 134°43'30″ = 0.79901993346 rad
∠ C' = γ' = 68.83215511542° = 68°49'54″ = 1.94402554567 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 = 9 ; ; b = 16 ; ; c = 21 ; ;

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

p = a+b+c = 9+16+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 * (23-9)(23-16)(23-21) } ; ; T = sqrt{ 4508 } = 67.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 67.14 }{ 9 } = 14.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 67.14 }{ 16 } = 8.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 67.14 }{ 21 } = 6.39 ; ;

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{ 9**2-16**2-21**2 }{ 2 * 16 * 21 } ) = 23° 33'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-9**2-21**2 }{ 2 * 9 * 21 } ) = 45° 16'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-9**2-16**2 }{ 2 * 16 * 9 } ) = 111° 10'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 67.14 }{ 23 } = 2.92 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 23° 33'23" } = 11.26 ; ;




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