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: h_a = 14.31663779527
Height: h_b = 7.49990551181
Height: h_c = 5.24993385827

Median: m_a = 25.30331618578
Median: m_b = 20.00662490237
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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