11 21 30 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 21   c = 30

Area: T = 78.74400787401
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 14.47656484197° = 14°28'32″ = 0.25326477263 rad
Angle ∠ B = β = 28.50435101171° = 28°30'13″ = 0.49774800999 rad
Angle ∠ C = γ = 137.0210841463° = 137°1'15″ = 2.39114648274 rad

Height: ha = 14.31663779527
Height: hb = 7.49990551181
Height: hc = 5.24993385827

Median: ma = 25.30331618578
Median: mb = 20.00662490237
Median: mc = 7.48333147735

Inradius: r = 2.544000254
Circumradius: R = 22.00327720028

Vertex coordinates: A[30; 0] B[0; 0] C[9.66766666667; 5.24993385827]
Centroid: CG[13.22222222222; 1.75497795276]
Coordinates of the circumscribed circle: U[15; -16.09772660973]
Coordinates of the inscribed circle: I[10; 2.544000254]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.524435158° = 165°31'28″ = 0.25326477263 rad
∠ B' = β' = 151.4966489883° = 151°29'47″ = 0.49774800999 rad
∠ C' = γ' = 42.97991585368° = 42°58'45″ = 2.39114648274 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 = 21 ; ; c = 30 ; ;

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

p = a+b+c = 11+21+30 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-11)(31-21)(31-30) } ; ; T = sqrt{ 6200 } = 78.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 78.74 }{ 11 } = 14.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 78.74 }{ 21 } = 7.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 78.74 }{ 30 } = 5.25 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 21**2+30**2-11**2 }{ 2 * 21 * 30 } ) = 14° 28'32" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11**2+30**2-21**2 }{ 2 * 11 * 30 } ) = 28° 30'13" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 11**2+21**2-30**2 }{ 2 * 11 * 21 } ) = 137° 1'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 78.74 }{ 31 } = 2.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 14° 28'32" } = 22 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 30**2 - 11**2 } }{ 2 } = 25.303 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 11**2 - 21**2 } }{ 2 } = 20.006 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 11**2 - 30**2 } }{ 2 } = 7.483 ; ;
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