12 15 20 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 15   c = 20

Area: T = 89.66656985697
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 36.71104466683° = 36°42'38″ = 0.64107181642 rad
Angle ∠ B = β = 48.35496321995° = 48°20'59″ = 0.8443860274 rad
Angle ∠ C = γ = 94.94399211322° = 94°56'24″ = 1.65770142153 rad

Height: ha = 14.9444283095
Height: hb = 11.9555426476
Height: hc = 8.9676569857

Median: ma = 16.62882891483
Median: mb = 14.68884308216
Median: mc = 9.19223881554

Inradius: r = 3.81655616413
Circumradius: R = 10.03772830899

Vertex coordinates: A[20; 0] B[0; 0] C[7.975; 8.9676569857]
Centroid: CG[9.325; 2.9898856619]
Coordinates of the circumscribed circle: U[10; -0.86443215994]
Coordinates of the inscribed circle: I[8.5; 3.81655616413]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.2989553332° = 143°17'22″ = 0.64107181642 rad
∠ B' = β' = 131.65503678° = 131°39'1″ = 0.8443860274 rad
∠ C' = γ' = 85.06600788678° = 85°3'36″ = 1.65770142153 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 = 20 ; ;

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

p = a+b+c = 12+15+20 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-12)(23.5-15)(23.5-20) } ; ; T = sqrt{ 8039.94 } = 89.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 89.67 }{ 12 } = 14.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 89.67 }{ 15 } = 11.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 89.67 }{ 20 } = 8.97 ; ;

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**2-20**2 }{ 2 * 15 * 20 } ) = 36° 42'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-12**2-20**2 }{ 2 * 12 * 20 } ) = 48° 20'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-12**2-15**2 }{ 2 * 15 * 12 } ) = 94° 56'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 89.67 }{ 23.5 } = 3.82 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 36° 42'38" } = 10.04 ; ;




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