40 50 80 triangle

Obtuse scalene triangle.

Sides: a = 40   b = 50   c = 80

Area: T = 818.1533408598
Perimeter: p = 170
Semiperimeter: s = 85

Angle ∠ A = α = 24.14768479965° = 24°8'49″ = 0.42114420015 rad
Angle ∠ B = β = 30.75435198081° = 30°45'13″ = 0.53767501772 rad
Angle ∠ C = γ = 125.1099632195° = 125°5'59″ = 2.18334004748 rad

Height: ha = 40.90876704299
Height: hb = 32.72661363439
Height: hc = 20.45438352149

Median: ma = 63.64396103068
Median: mb = 58.09547501931
Median: mc = 21.21332034356

Inradius: r = 9.62553342188
Circumradius: R = 48.89105865082

Vertex coordinates: A[80; 0] B[0; 0] C[34.375; 20.45438352149]
Centroid: CG[38.125; 6.81879450716]
Coordinates of the circumscribed circle: U[40; -28.11220872422]
Coordinates of the inscribed circle: I[35; 9.62553342188]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.8533152003° = 155°51'11″ = 0.42114420015 rad
∠ B' = β' = 149.2466480192° = 149°14'47″ = 0.53767501772 rad
∠ C' = γ' = 54.99003678046° = 54°54'1″ = 2.18334004748 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 = 40 ; ; b = 50 ; ; c = 80 ; ;

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

p = a+b+c = 40+50+80 = 170 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 170 }{ 2 } = 85 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 85 * (85-40)(85-50)(85-80) } ; ; T = sqrt{ 669375 } = 818.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 818.15 }{ 40 } = 40.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 818.15 }{ 50 } = 32.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 818.15 }{ 80 } = 20.45 ; ;

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{ 50**2+80**2-40**2 }{ 2 * 50 * 80 } ) = 24° 8'49" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40**2+80**2-50**2 }{ 2 * 40 * 80 } ) = 30° 45'13" ; ; gamma = 180° - alpha - beta = 180° - 24° 8'49" - 30° 45'13" = 125° 5'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 818.15 }{ 85 } = 9.63 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 40 }{ 2 * sin 24° 8'49" } = 48.89 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 80**2 - 40**2 } }{ 2 } = 63.64 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 40**2 - 50**2 } }{ 2 } = 58.095 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 40**2 - 80**2 } }{ 2 } = 21.213 ; ;
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