45 32 31 triangle

Obtuse scalene triangle.

Sides: a = 45   b = 32   c = 31

Area: T = 495.8999183302
Perimeter: p = 108
Semiperimeter: s = 54

Angle ∠ A = α = 91.15552351169° = 91°9'19″ = 1.59109589832 rad
Angle ∠ B = β = 45.31436127157° = 45°18'49″ = 0.79108717379 rad
Angle ∠ C = γ = 43.53111521674° = 43°31'52″ = 0.76597619325 rad

Height: ha = 22.04399637023
Height: hb = 30.99436989564
Height: hc = 31.99334956969

Median: ma = 22.05110770712
Median: mb = 35.17110107901
Median: mc = 35.83664339744

Inradius: r = 9.18333182093
Circumradius: R = 22.50545742679

Vertex coordinates: A[31; 0] B[0; 0] C[31.64551612903; 31.99334956969]
Centroid: CG[20.88217204301; 10.66444985656]
Coordinates of the circumscribed circle: U[15.5; 16.31658163442]
Coordinates of the inscribed circle: I[22; 9.18333182093]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 88.84547648831° = 88°50'41″ = 1.59109589832 rad
∠ B' = β' = 134.6866387284° = 134°41'11″ = 0.79108717379 rad
∠ C' = γ' = 136.4698847833° = 136°28'8″ = 0.76597619325 rad

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How did we calculate this triangle?

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

p = a+b+c = 45+32+31 = 108 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 108 }{ 2 } = 54 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54 * (54-45)(54-32)(54-31) } ; ; T = sqrt{ 245916 } = 495.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 495.9 }{ 45 } = 22.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 495.9 }{ 32 } = 30.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 495.9 }{ 31 } = 31.99 ; ;

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{ 32**2+31**2-45**2 }{ 2 * 32 * 31 } ) = 91° 9'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+31**2-32**2 }{ 2 * 45 * 31 } ) = 45° 18'49" ; ; gamma = 180° - alpha - beta = 180° - 91° 9'19" - 45° 18'49" = 43° 31'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 495.9 }{ 54 } = 9.18 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 91° 9'19" } = 22.5 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 31**2 - 45**2 } }{ 2 } = 22.051 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 45**2 - 32**2 } }{ 2 } = 35.171 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 45**2 - 31**2 } }{ 2 } = 35.836 ; ;
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