11 14 21 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 14   c = 21

Area: T = 70.48440407468
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 28.6521659028° = 28°39'6″ = 0.55000657862 rad
Angle ∠ B = β = 37.60876855758° = 37°36'28″ = 0.65663779374 rad
Angle ∠ C = γ = 113.7410655396° = 113°44'26″ = 1.985514893 rad

Height: ha = 12.81552801358
Height: hb = 10.06991486781
Height: hc = 6.71327657854

Median: ma = 16.97879268463
Median: mb = 15.23215462117
Median: mc = 6.94662219947

Inradius: r = 3.06545235107
Circumradius: R = 11.4710681752

Vertex coordinates: A[21; 0] B[0; 0] C[8.71442857143; 6.71327657854]
Centroid: CG[9.90547619048; 2.23875885951]
Coordinates of the circumscribed circle: U[10.5; -4.61880666794]
Coordinates of the inscribed circle: I[9; 3.06545235107]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.3488340972° = 151°20'54″ = 0.55000657862 rad
∠ B' = β' = 142.3922314424° = 142°23'32″ = 0.65663779374 rad
∠ C' = γ' = 66.25993446038° = 66°15'34″ = 1.985514893 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 = 11 ; ; b = 14 ; ; c = 21 ; ;

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

p = a+b+c = 11+14+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 * (23-11)(23-14)(23-21) } ; ; T = sqrt{ 4968 } = 70.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 70.48 }{ 11 } = 12.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 70.48 }{ 14 } = 10.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 70.48 }{ 21 } = 6.71 ; ;

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{ 11**2-14**2-21**2 }{ 2 * 14 * 21 } ) = 28° 39'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-11**2-21**2 }{ 2 * 11 * 21 } ) = 37° 36'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-11**2-14**2 }{ 2 * 14 * 11 } ) = 113° 44'26" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 70.48 }{ 23 } = 3.06 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 28° 39'6" } = 11.47 ; ;




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