35 60 90 triangle

Obtuse scalene triangle.

Sides: a = 35   b = 60   c = 90

Area: T = 657.387997954
Perimeter: p = 185
Semiperimeter: s = 92.5

Angle ∠ A = α = 14.09216737232° = 14°5'30″ = 0.24659461036 rad
Angle ∠ B = β = 24.67695454829° = 24°40'10″ = 0.43105647936 rad
Angle ∠ C = γ = 141.2398780794° = 141°14'20″ = 2.46550817564 rad

Height: ha = 37.56545702594
Height: hb = 21.91326659847
Height: hc = 14.60884439898

Median: ma = 74.45663630592
Median: mb = 61.33992207319
Median: mc = 19.6855019685

Inradius: r = 7.10768105896
Circumradius: R = 71.87662382041

Vertex coordinates: A[90; 0] B[0; 0] C[31.80655555556; 14.60884439898]
Centroid: CG[40.60218518519; 4.86994813299]
Coordinates of the circumscribed circle: U[45; -56.04663524091]
Coordinates of the inscribed circle: I[32.5; 7.10768105896]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.9088326277° = 165°54'30″ = 0.24659461036 rad
∠ B' = β' = 155.3330454517° = 155°19'50″ = 0.43105647936 rad
∠ C' = γ' = 38.76112192061° = 38°45'40″ = 2.46550817564 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 = 35 ; ; b = 60 ; ; c = 90 ; ;

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

p = a+b+c = 35+60+90 = 185 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 185 }{ 2 } = 92.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 92.5 * (92.5-35)(92.5-60)(92.5-90) } ; ; T = sqrt{ 432148.44 } = 657.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 657.38 }{ 35 } = 37.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 657.38 }{ 60 } = 21.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 657.38 }{ 90 } = 14.61 ; ;

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{ 60**2+90**2-35**2 }{ 2 * 60 * 90 } ) = 14° 5'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+90**2-60**2 }{ 2 * 35 * 90 } ) = 24° 40'10" ; ; gamma = 180° - alpha - beta = 180° - 14° 5'30" - 24° 40'10" = 141° 14'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 657.38 }{ 92.5 } = 7.11 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 14° 5'30" } = 71.88 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 90**2 - 35**2 } }{ 2 } = 74.456 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 35**2 - 60**2 } }{ 2 } = 61.339 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 35**2 - 90**2 } }{ 2 } = 19.685 ; ;
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