40 50 60 triangle

Acute scalene triangle.

Sides: a = 40   b = 50   c = 60

Area: T = 992.1576741649
Perimeter: p = 150
Semiperimeter: s = 75

Angle ∠ A = α = 41.41096221093° = 41°24'35″ = 0.72327342478 rad
Angle ∠ B = β = 55.77111336722° = 55°46'16″ = 0.97333899101 rad
Angle ∠ C = γ = 82.81992442185° = 82°49'9″ = 1.44554684956 rad

Height: ha = 49.60878370825
Height: hb = 39.6866269666
Height: hc = 33.07218913883

Median: ma = 51.47881507049
Median: mb = 44.44109720866
Median: mc = 33.91216499156

Inradius: r = 13.22987565553
Circumradius: R = 30.23771578407

Vertex coordinates: A[60; 0] B[0; 0] C[22.5; 33.07218913883]
Centroid: CG[27.5; 11.02439637961]
Coordinates of the circumscribed circle: U[30; 3.78796447301]
Coordinates of the inscribed circle: I[25; 13.22987565553]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.5990377891° = 138°35'25″ = 0.72327342478 rad
∠ B' = β' = 124.2298866328° = 124°13'44″ = 0.97333899101 rad
∠ C' = γ' = 97.18107557815° = 97°10'51″ = 1.44554684956 rad

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How did we calculate this triangle?

a = 40 ; ; b = 50 ; ; c = 60 ; ;

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

p = a+b+c = 40+50+60 = 150 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 150 }{ 2 } = 75 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 75 * (75-40)(75-50)(75-60) } ; ; T = sqrt{ 984375 } = 992.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 992.16 }{ 40 } = 49.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 992.16 }{ 50 } = 39.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 992.16 }{ 60 } = 33.07 ; ;

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{ 40**2-50**2-60**2 }{ 2 * 50 * 60 } ) = 41° 24'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 50**2-40**2-60**2 }{ 2 * 40 * 60 } ) = 55° 46'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 60**2-40**2-50**2 }{ 2 * 50 * 40 } ) = 82° 49'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 992.16 }{ 75 } = 13.23 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 40 }{ 2 * sin 41° 24'35" } = 30.24 ; ;




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