9 18 24 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 18   c = 24

Area: T = 68.87999818314
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 18.57333497187° = 18°34'24″ = 0.32441661057 rad
Angle ∠ B = β = 39.57112194572° = 39°34'16″ = 0.69106480686 rad
Angle ∠ C = γ = 121.8555430824° = 121°51'20″ = 2.12767784793 rad

Height: ha = 15.28988848514
Height: hb = 7.64444424257
Height: hc = 5.73333318193

Median: ma = 20.73304124416
Median: mb = 15.73221327226
Median: mc = 7.64985292704

Inradius: r = 2.69880385032
Circumradius: R = 14.12879107076

Vertex coordinates: A[24; 0] B[0; 0] C[6.93875; 5.73333318193]
Centroid: CG[10.31325; 1.91111106064]
Coordinates of the circumscribed circle: U[12; -7.45663973179]
Coordinates of the inscribed circle: I[7.5; 2.69880385032]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.4276650281° = 161°25'36″ = 0.32441661057 rad
∠ B' = β' = 140.4298780543° = 140°25'44″ = 0.69106480686 rad
∠ C' = γ' = 58.1454569176° = 58°8'40″ = 2.12767784793 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 = 18 ; ; c = 24 ; ;

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

p = a+b+c = 9+18+24 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-9)(25.5-18)(25.5-24) } ; ; T = sqrt{ 4733.44 } = 68.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 68.8 }{ 9 } = 15.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 68.8 }{ 18 } = 7.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 68.8 }{ 24 } = 5.73 ; ;

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**2-18**2-24**2 }{ 2 * 18 * 24 } ) = 18° 34'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-9**2-24**2 }{ 2 * 9 * 24 } ) = 39° 34'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-9**2-18**2 }{ 2 * 18 * 9 } ) = 121° 51'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 68.8 }{ 25.5 } = 2.7 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 18° 34'24" } = 14.13 ; ;




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