12 21 30 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 21   c = 30

Area: T = 98.35987184748
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 18.19548723388° = 18°11'42″ = 0.31875604293 rad
Angle ∠ B = β = 33.12329402077° = 33°7'23″ = 0.57881043646 rad
Angle ∠ C = γ = 128.6822187453° = 128°40'56″ = 2.24659278597 rad

Height: ha = 16.39331197458
Height: hb = 9.36774969976
Height: hc = 6.55772478983

Median: ma = 25.18992834356
Median: mb = 20.29216238877
Median: mc = 8.21658383626

Inradius: r = 3.12224989992
Circumradius: R = 19.21553784566

Vertex coordinates: A[30; 0] B[0; 0] C[10.05; 6.55772478983]
Centroid: CG[13.35; 2.18657492994]
Coordinates of the circumscribed circle: U[15; -12.01096115354]
Coordinates of the inscribed circle: I[10.5; 3.12224989992]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.8055127661° = 161°48'18″ = 0.31875604293 rad
∠ B' = β' = 146.8777059792° = 146°52'37″ = 0.57881043646 rad
∠ C' = γ' = 51.31878125465° = 51°19'4″ = 2.24659278597 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 = 12 ; ; b = 21 ; ; c = 30 ; ;

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

p = a+b+c = 12+21+30 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-12)(31.5-21)(31.5-30) } ; ; T = sqrt{ 9674.44 } = 98.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 98.36 }{ 12 } = 16.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 98.36 }{ 21 } = 9.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 98.36 }{ 30 } = 6.56 ; ;

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{ 12**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 18° 11'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-12**2-30**2 }{ 2 * 12 * 30 } ) = 33° 7'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-12**2-21**2 }{ 2 * 21 * 12 } ) = 128° 40'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 98.36 }{ 31.5 } = 3.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 18° 11'42" } = 19.22 ; ;




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