Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Obtuse scalene triangle.

Sides: a = 68   b = 110   c = 75.46107914173

Area: T = 2526.707737648
Perimeter: p = 253.4610791417
Semiperimeter: s = 126.7330395709

Angle ∠ A = α = 37.50224069243° = 37°30'9″ = 0.65545404783 rad
Angle ∠ B = β = 99.99875930757° = 99°59'51″ = 1.74552872432 rad
Angle ∠ C = γ = 42.5° = 42°30' = 0.74217649321 rad

Height: ha = 74.31549228377
Height: hb = 45.94401341179
Height: hc = 66.96774231883

Median: ma = 87.98438935298
Median: mb = 46.19770293489
Median: mc = 83.29771622546

Inradius: r = 19.93876586994
Circumradius: R = 55.8488050021

Vertex coordinates: A[75.46107914173; 0] B[0; 0] C[-11.80552628737; 66.96774231883]
Centroid: CG[21.21985095145; 22.32224743961]
Coordinates of the circumscribed circle: U[37.73303957086; 41.17655015855]
Coordinates of the inscribed circle: I[16.73303957086; 19.93876586994]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.4987593076° = 142°29'51″ = 0.65545404783 rad
∠ B' = β' = 80.00224069243° = 80°9″ = 1.74552872432 rad
∠ C' = γ' = 137.5° = 137°30' = 0.74217649321 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

a = 68 ; ; b = 110 ; ; gamma = 42° 30' ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 68**2+110**2 - 2 * 68 * 110 * cos 42° 30' } ; ; c = 75.46 ; ;


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 = 68 ; ; b = 110 ; ; c = 75.46 ; ;

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

p = a+b+c = 68+110+75.46 = 253.46 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 253.46 }{ 2 } = 126.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 126.73 * (126.73-68)(126.73-110)(126.73-75.46) } ; ; T = sqrt{ 6384250.17 } = 2526.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2526.71 }{ 68 } = 74.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2526.71 }{ 110 } = 45.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2526.71 }{ 75.46 } = 66.97 ; ;

6. 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{ 110**2+75.46**2-68**2 }{ 2 * 110 * 75.46 } ) = 37° 30'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 68**2+75.46**2-110**2 }{ 2 * 68 * 75.46 } ) = 99° 59'51" ; ; gamma = 180° - alpha - beta = 180° - 37° 30'9" - 99° 59'51" = 42° 30' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2526.71 }{ 126.73 } = 19.94 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 68 }{ 2 * sin 37° 30'9" } = 55.85 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 110**2+2 * 75.46**2 - 68**2 } }{ 2 } = 87.984 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 75.46**2+2 * 68**2 - 110**2 } }{ 2 } = 46.197 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 110**2+2 * 68**2 - 75.46**2 } }{ 2 } = 83.297 ; ;
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