9 12 16 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 12   c = 16

Area: T = 53.44109721094
Perimeter: p = 37
Semiperimeter: s = 18.5

Angle ∠ A = α = 33.82662859376° = 33°49'35″ = 0.59903800633 rad
Angle ∠ B = β = 47.92222147824° = 47°55'20″ = 0.83664004328 rad
Angle ∠ C = γ = 98.251149928° = 98°15'5″ = 1.71548121575 rad

Height: ha = 11.87657715799
Height: hb = 8.90768286849
Height: hc = 6.68801215137

Median: ma = 13.40770876778
Median: mb = 11.51108644332
Median: mc = 6.96441941386

Inradius: r = 2.88987011951
Circumradius: R = 8.08436852877

Vertex coordinates: A[16; 0] B[0; 0] C[6.031125; 6.68801215137]
Centroid: CG[7.344375; 2.22767071712]
Coordinates of the circumscribed circle: U[8; -1.16601585367]
Coordinates of the inscribed circle: I[6.5; 2.88987011951]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.1743714062° = 146°10'25″ = 0.59903800633 rad
∠ B' = β' = 132.0787785218° = 132°4'40″ = 0.83664004328 rad
∠ C' = γ' = 81.749850072° = 81°44'55″ = 1.71548121575 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 = 12 ; ; c = 16 ; ;

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

p = a+b+c = 9+12+16 = 37 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37 }{ 2 } = 18.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.5 * (18.5-9)(18.5-12)(18.5-16) } ; ; T = sqrt{ 2855.94 } = 53.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.44 }{ 9 } = 11.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.44 }{ 12 } = 8.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.44 }{ 16 } = 6.68 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12**2+16**2-9**2 }{ 2 * 12 * 16 } ) = 33° 49'35" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+16**2-12**2 }{ 2 * 9 * 16 } ) = 47° 55'20" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 9**2+12**2-16**2 }{ 2 * 9 * 12 } ) = 98° 15'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.44 }{ 18.5 } = 2.89 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 33° 49'35" } = 8.08 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 16**2 - 9**2 } }{ 2 } = 13.407 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16**2+2 * 9**2 - 12**2 } }{ 2 } = 11.511 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 9**2 - 16**2 } }{ 2 } = 6.964 ; ;
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