19 29 30 triangle

Acute scalene triangle.

Sides: a = 19   b = 29   c = 30

Area: T = 264.953282599
Perimeter: p = 78
Semiperimeter: s = 39

Angle ∠ A = α = 37.52435132021° = 37°31'25″ = 0.65549088523 rad
Angle ∠ B = β = 68.38217275778° = 68°22'54″ = 1.19334862944 rad
Angle ∠ C = γ = 74.095475922° = 74°5'41″ = 1.29331975069 rad

Height: ha = 27.89897711568
Height: hb = 18.2732608689
Height: hc = 17.66435217327

Median: ma = 27.93329554469
Median: mb = 20.5
Median: mc = 19.39107194297

Inradius: r = 6.79436622049
Circumradius: R = 15.59771161453

Vertex coordinates: A[30; 0] B[0; 0] C[7; 17.66435217327]
Centroid: CG[12.33333333333; 5.88878405776]
Coordinates of the circumscribed circle: U[15; 4.27443458039]
Coordinates of the inscribed circle: I[10; 6.79436622049]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.4766486798° = 142°28'35″ = 0.65549088523 rad
∠ B' = β' = 111.6188272422° = 111°37'6″ = 1.19334862944 rad
∠ C' = γ' = 105.905524078° = 105°54'19″ = 1.29331975069 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 = 19 ; ; b = 29 ; ; c = 30 ; ;

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

p = a+b+c = 19+29+30 = 78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78 }{ 2 } = 39 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39 * (39-19)(39-29)(39-30) } ; ; T = sqrt{ 70200 } = 264.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 264.95 }{ 19 } = 27.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 264.95 }{ 29 } = 18.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 264.95 }{ 30 } = 17.66 ; ;

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{ 19**2-29**2-30**2 }{ 2 * 29 * 30 } ) = 37° 31'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29**2-19**2-30**2 }{ 2 * 19 * 30 } ) = 68° 22'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-19**2-29**2 }{ 2 * 29 * 19 } ) = 74° 5'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 264.95 }{ 39 } = 6.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 37° 31'25" } = 15.6 ; ;




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