11 15 21 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 15   c = 21

Area: T = 79.00875154653
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 30.10882310581° = 30°6'30″ = 0.52554877639 rad
Angle ∠ B = β = 43.1610764259° = 43°9'39″ = 0.7533297444 rad
Angle ∠ C = γ = 106.7311004683° = 106°43'52″ = 1.86328074457 rad

Height: ha = 14.36550028119
Height: hb = 10.53443353954
Height: hc = 7.52545252824

Median: ma = 17.43997126413
Median: mb = 14.99216643506
Median: mc = 7.92114897589

Inradius: r = 3.36220219347
Circumradius: R = 10.9644146827

Vertex coordinates: A[21; 0] B[0; 0] C[8.02438095238; 7.52545252824]
Centroid: CG[9.67546031746; 2.50881750941]
Coordinates of the circumscribed circle: U[10.5; -3.15663452987]
Coordinates of the inscribed circle: I[8.5; 3.36220219347]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.8921768942° = 149°53'30″ = 0.52554877639 rad
∠ B' = β' = 136.8399235741° = 136°50'21″ = 0.7533297444 rad
∠ C' = γ' = 73.2698995317° = 73°16'8″ = 1.86328074457 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 = 15 ; ; c = 21 ; ;

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

p = a+b+c = 11+15+21 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-11)(23.5-15)(23.5-21) } ; ; T = sqrt{ 6242.19 } = 79.01 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 79.01 }{ 11 } = 14.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 79.01 }{ 15 } = 10.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 79.01 }{ 21 } = 7.52 ; ;

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**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 30° 6'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-11**2-21**2 }{ 2 * 11 * 21 } ) = 43° 9'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-11**2-15**2 }{ 2 * 15 * 11 } ) = 106° 43'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 79.01 }{ 23.5 } = 3.36 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 30° 6'30" } = 10.96 ; ;




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