43 80 56 triangle

Obtuse scalene triangle.

Sides: a = 43   b = 80   c = 56

Area: T = 1150.859921706
Perimeter: p = 179
Semiperimeter: s = 89.5

Angle ∠ A = α = 30.91657052964° = 30°54'57″ = 0.5439580848 rad
Angle ∠ B = β = 107.0866330888° = 107°5'11″ = 1.86990090579 rad
Angle ∠ C = γ = 41.99879638158° = 41°59'53″ = 0.73330027477 rad

Height: ha = 53.52883356774
Height: hb = 28.77114804266
Height: hc = 41.10221148951

Median: ma = 65.61882139349
Median: mb = 29.87547384926
Median: mc = 57.79770587487

Inradius: r = 12.85987622018
Circumradius: R = 41.8476995085

Vertex coordinates: A[56; 0] B[0; 0] C[-12.63439285714; 41.10221148951]
Centroid: CG[14.45553571429; 13.7010704965]
Coordinates of the circumscribed circle: U[28; 31.09993729462]
Coordinates of the inscribed circle: I[9.5; 12.85987622018]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.0844294704° = 149°5'3″ = 0.5439580848 rad
∠ B' = β' = 72.91436691122° = 72°54'49″ = 1.86990090579 rad
∠ C' = γ' = 138.0022036184° = 138°7″ = 0.73330027477 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 = 43 ; ; b = 80 ; ; c = 56 ; ;

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

p = a+b+c = 43+80+56 = 179 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 179 }{ 2 } = 89.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 89.5 * (89.5-43)(89.5-80)(89.5-56) } ; ; T = sqrt{ 1324476.94 } = 1150.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1150.86 }{ 43 } = 53.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1150.86 }{ 80 } = 28.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1150.86 }{ 56 } = 41.1 ; ;

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{ 80**2+56**2-43**2 }{ 2 * 80 * 56 } ) = 30° 54'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 43**2+56**2-80**2 }{ 2 * 43 * 56 } ) = 107° 5'11" ; ;
 gamma = 180° - alpha - beta = 180° - 30° 54'57" - 107° 5'11" = 41° 59'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1150.86 }{ 89.5 } = 12.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 43 }{ 2 * sin 30° 54'57" } = 41.85 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 56**2 - 43**2 } }{ 2 } = 65.618 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 43**2 - 80**2 } }{ 2 } = 29.875 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 43**2 - 56**2 } }{ 2 } = 57.797 ; ;
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