15 19 24 triangle

Acute scalene triangle.

Sides: a = 15   b = 19   c = 24

Area: T = 142.4788068488
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 38.67551276498° = 38°40'30″ = 0.67550083161 rad
Angle ∠ B = β = 52.33301130357° = 52°19'48″ = 0.91333327704 rad
Angle ∠ C = γ = 88.99547593146° = 88°59'41″ = 1.55332515671 rad

Height: ha = 18.99770757984
Height: hb = 14.99876914198
Height: hc = 11.8733172374

Median: ma = 20.30439405042
Median: mb = 17.61439149538
Median: mc = 12.20765556157

Inradius: r = 4.91330368444
Circumradius: R = 12.00218471485

Vertex coordinates: A[24; 0] B[0; 0] C[9.16766666667; 11.8733172374]
Centroid: CG[11.05655555556; 3.95877241247]
Coordinates of the circumscribed circle: U[12; 0.21105587219]
Coordinates of the inscribed circle: I[10; 4.91330368444]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.325487235° = 141°19'30″ = 0.67550083161 rad
∠ B' = β' = 127.6769886964° = 127°40'12″ = 0.91333327704 rad
∠ C' = γ' = 91.00552406854° = 91°19″ = 1.55332515671 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 = 15 ; ; b = 19 ; ; c = 24 ; ;

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

p = a+b+c = 15+19+24 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-15)(29-19)(29-24) } ; ; T = sqrt{ 20300 } = 142.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 142.48 }{ 15 } = 19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 142.48 }{ 19 } = 15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 142.48 }{ 24 } = 11.87 ; ;

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{ 15**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 38° 40'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-15**2-24**2 }{ 2 * 15 * 24 } ) = 52° 19'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-15**2-19**2 }{ 2 * 19 * 15 } ) = 88° 59'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 142.48 }{ 29 } = 4.91 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 38° 40'30" } = 12 ; ;




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