10 19 21 triangle

Acute scalene triangle.

Sides: a = 10   b = 19   c = 21

Area: T = 94.86883298051
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 28.3943894835° = 28°23'38″ = 0.49655669523 rad
Angle ∠ B = β = 64.62330664748° = 64°37'23″ = 1.12878852827 rad
Angle ∠ C = γ = 86.98330386902° = 86°58'59″ = 1.51881404185 rad

Height: ha = 18.9743665961
Height: hb = 9.98661399795
Height: hc = 9.03550790291

Median: ma = 19.39107194297
Median: mb = 13.42657215821
Median: mc = 10.96658560997

Inradius: r = 3.79547331922
Circumradius: R = 10.51545732201

Vertex coordinates: A[21; 0] B[0; 0] C[4.28657142857; 9.03550790291]
Centroid: CG[8.42985714286; 3.01216930097]
Coordinates of the circumscribed circle: U[10.5; 0.55333985905]
Coordinates of the inscribed circle: I[6; 3.79547331922]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.6066105165° = 151°36'22″ = 0.49655669523 rad
∠ B' = β' = 115.3776933525° = 115°22'37″ = 1.12878852827 rad
∠ C' = γ' = 93.01769613098° = 93°1'1″ = 1.51881404185 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 = 10 ; ; b = 19 ; ; c = 21 ; ;

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

p = a+b+c = 10+19+21 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-10)(25-19)(25-21) } ; ; T = sqrt{ 9000 } = 94.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 94.87 }{ 10 } = 18.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 94.87 }{ 19 } = 9.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 94.87 }{ 21 } = 9.04 ; ;

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{ 19**2+21**2-10**2 }{ 2 * 19 * 21 } ) = 28° 23'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+21**2-19**2 }{ 2 * 10 * 21 } ) = 64° 37'23" ; ; gamma = 180° - alpha - beta = 180° - 28° 23'38" - 64° 37'23" = 86° 58'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 94.87 }{ 25 } = 3.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 28° 23'38" } = 10.51 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 19**2+2 * 21**2 - 10**2 } }{ 2 } = 19.391 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 10**2 - 19**2 } }{ 2 } = 13.426 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 19**2+2 * 10**2 - 21**2 } }{ 2 } = 10.966 ; ;
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