9 24 30 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 24   c = 30

Area: T = 89.29441067484
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 14.36215115629° = 14°21'41″ = 0.25106556623 rad
Angle ∠ B = β = 41.41096221093° = 41°24'35″ = 0.72327342478 rad
Angle ∠ C = γ = 124.2298866328° = 124°13'44″ = 2.16882027434 rad

Height: ha = 19.8433134833
Height: hb = 7.44111755624
Height: hc = 5.95329404499

Median: ma = 26.79108566492
Median: mb = 18.6154510469
Median: mc = 10.17334949747

Inradius: r = 2.83547335476
Circumradius: R = 18.14222947044

Vertex coordinates: A[30; 0] B[0; 0] C[6.75; 5.95329404499]
Centroid: CG[12.25; 1.98443134833]
Coordinates of the circumscribed circle: U[15; -10.20550407712]
Coordinates of the inscribed circle: I[7.5; 2.83547335476]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.6388488437° = 165°38'19″ = 0.25106556623 rad
∠ B' = β' = 138.5990377891° = 138°35'25″ = 0.72327342478 rad
∠ C' = γ' = 55.77111336722° = 55°46'16″ = 2.16882027434 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 = 9 ; ; b = 24 ; ; c = 30 ; ;

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

p = a+b+c = 9+24+30 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-9)(31.5-24)(31.5-30) } ; ; T = sqrt{ 7973.44 } = 89.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 89.29 }{ 9 } = 19.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 89.29 }{ 24 } = 7.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 89.29 }{ 30 } = 5.95 ; ;

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{ 9**2-24**2-30**2 }{ 2 * 24 * 30 } ) = 14° 21'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-9**2-30**2 }{ 2 * 9 * 30 } ) = 41° 24'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-9**2-24**2 }{ 2 * 24 * 9 } ) = 124° 13'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 89.29 }{ 31.5 } = 2.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 14° 21'41" } = 18.14 ; ;




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