13 19 24 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 19   c = 24

Area: T = 122.9633409192
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 32.63768975036° = 32°38'13″ = 0.57696213191 rad
Angle ∠ B = β = 52.02201275551° = 52°1'12″ = 0.90879225031 rad
Angle ∠ C = γ = 95.34329749413° = 95°20'35″ = 1.66440488314 rad

Height: ha = 18.91774475679
Height: hb = 12.9443516757
Height: hc = 10.2476950766

Median: ma = 20.64658228221
Median: mb = 16.88002976164
Median: mc = 11

Inradius: r = 4.39215503283
Circumradius: R = 12.05223659009

Vertex coordinates: A[24; 0] B[0; 0] C[8; 10.2476950766]
Centroid: CG[10.66766666667; 3.41656502553]
Coordinates of the circumscribed circle: U[12; -1.12222850839]
Coordinates of the inscribed circle: I[9; 4.39215503283]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.3633102496° = 147°21'47″ = 0.57696213191 rad
∠ B' = β' = 127.9879872445° = 127°58'48″ = 0.90879225031 rad
∠ C' = γ' = 84.65770250587° = 84°39'25″ = 1.66440488314 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 = 13 ; ; b = 19 ; ; c = 24 ; ;

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

p = a+b+c = 13+19+24 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-13)(28-19)(28-24) } ; ; T = sqrt{ 15120 } = 122.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 122.96 }{ 13 } = 18.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 122.96 }{ 19 } = 12.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 122.96 }{ 24 } = 10.25 ; ;

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{ 13**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 32° 38'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-13**2-24**2 }{ 2 * 13 * 24 } ) = 52° 1'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-13**2-19**2 }{ 2 * 19 * 13 } ) = 95° 20'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 122.96 }{ 28 } = 4.39 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 32° 38'13" } = 12.05 ; ;




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