12 29 30 triangle

Acute scalene triangle.

Sides: a = 12   b = 29   c = 30

Area: T = 172.698753183
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 23.39111307904° = 23°23'28″ = 0.40882522481 rad
Angle ∠ B = β = 73.62437104336° = 73°37'25″ = 1.28549761546 rad
Angle ∠ C = γ = 82.98551587759° = 82°59'7″ = 1.44883642509 rad

Height: ha = 28.78329219716
Height: hb = 11.91101746089
Height: hc = 11.51331687886

Median: ma = 28.88877136513
Median: mb = 17.6566443583
Median: mc = 16.35554272338

Inradius: r = 4.86547192065
Circumradius: R = 15.11331285569

Vertex coordinates: A[30; 0] B[0; 0] C[3.38333333333; 11.51331687886]
Centroid: CG[11.12877777778; 3.83877229295]
Coordinates of the circumscribed circle: U[15; 1.84657125393]
Coordinates of the inscribed circle: I[6.5; 4.86547192065]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.609886921° = 156°36'32″ = 0.40882522481 rad
∠ B' = β' = 106.3766289566° = 106°22'35″ = 1.28549761546 rad
∠ C' = γ' = 97.01548412241° = 97°53″ = 1.44883642509 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 = 29 ; ; c = 30 ; ;

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

p = a+b+c = 12+29+30 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-12)(35.5-29)(35.5-30) } ; ; T = sqrt{ 29824.44 } = 172.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 172.7 }{ 12 } = 28.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 172.7 }{ 29 } = 11.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 172.7 }{ 30 } = 11.51 ; ;

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**2-29**2-30**2 }{ 2 * 29 * 30 } ) = 23° 23'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29**2-12**2-30**2 }{ 2 * 12 * 30 } ) = 73° 37'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-12**2-29**2 }{ 2 * 29 * 12 } ) = 82° 59'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 172.7 }{ 35.5 } = 4.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 23° 23'28" } = 15.11 ; ;




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