19 24 30 triangle

Acute scalene triangle.

Sides: a = 19   b = 24   c = 30

Area: T = 227.8122285665
Perimeter: p = 73
Semiperimeter: s = 36.5

Angle ∠ A = α = 39.25878857954° = 39°15'28″ = 0.68551793645 rad
Angle ∠ B = β = 53.06772521683° = 53°4'2″ = 0.92661982753 rad
Angle ∠ C = γ = 87.67548620363° = 87°40'29″ = 1.53302150138 rad

Height: ha = 23.98802405963
Height: hb = 18.98443571388
Height: hc = 15.1877485711

Median: ma = 25.45109331852
Median: mb = 22.05767450001
Median: mc = 15.60444865343

Inradius: r = 6.2411432484
Circumradius: R = 15.01223598032

Vertex coordinates: A[30; 0] B[0; 0] C[11.41766666667; 15.1877485711]
Centroid: CG[13.80655555556; 5.0622495237]
Coordinates of the circumscribed circle: U[15; 0.6099054071]
Coordinates of the inscribed circle: I[12.5; 6.2411432484]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.7422114205° = 140°44'32″ = 0.68551793645 rad
∠ B' = β' = 126.9332747832° = 126°55'58″ = 0.92661982753 rad
∠ C' = γ' = 92.32551379637° = 92°19'31″ = 1.53302150138 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 = 24 ; ; c = 30 ; ;

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

p = a+b+c = 19+24+30 = 73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73 }{ 2 } = 36.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.5 * (36.5-19)(36.5-24)(36.5-30) } ; ; T = sqrt{ 51898.44 } = 227.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 227.81 }{ 19 } = 23.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 227.81 }{ 24 } = 18.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 227.81 }{ 30 } = 15.19 ; ;

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**2-24**2-30**2 }{ 2 * 24 * 30 } ) = 39° 15'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-19**2-30**2 }{ 2 * 19 * 30 } ) = 53° 4'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-19**2-24**2 }{ 2 * 24 * 19 } ) = 87° 40'29" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 227.81 }{ 36.5 } = 6.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 39° 15'28" } = 15.01 ; ;




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