12 15 24 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 15   c = 24

Area: T = 73.63438067738
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 24.14768479965° = 24°8'49″ = 0.42114420015 rad
Angle ∠ B = β = 30.75435198081° = 30°45'13″ = 0.53767501772 rad
Angle ∠ C = γ = 125.1099632195° = 125°5'59″ = 2.18334004748 rad

Height: ha = 12.2722301129
Height: hb = 9.81878409032
Height: hc = 6.13661505645

Median: ma = 19.0921883092
Median: mb = 17.42884250579
Median: mc = 6.36439610307

Inradius: r = 2.88876002656
Circumradius: R = 14.66771759525

Vertex coordinates: A[24; 0] B[0; 0] C[10.31325; 6.13661505645]
Centroid: CG[11.43875; 2.04553835215]
Coordinates of the circumscribed circle: U[12; -8.43436261727]
Coordinates of the inscribed circle: I[10.5; 2.88876002656]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.8533152003° = 155°51'11″ = 0.42114420015 rad
∠ B' = β' = 149.2466480192° = 149°14'47″ = 0.53767501772 rad
∠ C' = γ' = 54.99003678046° = 54°54'1″ = 2.18334004748 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 = 12 ; ; b = 15 ; ; c = 24 ; ;

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

p = a+b+c = 12+15+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 * (25.5-12)(25.5-15)(25.5-24) } ; ; T = sqrt{ 5421.94 } = 73.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 73.63 }{ 12 } = 12.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 73.63 }{ 15 } = 9.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 73.63 }{ 24 } = 6.14 ; ;

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{ 12**2-15**2-24**2 }{ 2 * 15 * 24 } ) = 24° 8'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-12**2-24**2 }{ 2 * 12 * 24 } ) = 30° 45'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-12**2-15**2 }{ 2 * 15 * 12 } ) = 125° 5'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 73.63 }{ 25.5 } = 2.89 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 24° 8'49" } = 14.67 ; ;




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