Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 84.49   b = 193.92   c = 212.2021514757

Area: T = 8191.838846801
Perimeter: p = 490.6121514757
Semiperimeter: s = 245.3065757378

Angle ∠ A = α = 23.46222088863° = 23°27'44″ = 0.40994927949 rad
Angle ∠ B = β = 66.03877911137° = 66°2'16″ = 1.15325768857 rad
Angle ∠ C = γ = 90.5° = 90°30' = 1.58795229731 rad

Height: ha = 193.9132616121
Height: hb = 84.48767828797
Height: hc = 77.20881054879

Median: ma = 198.8288027721
Median: mb = 129.1621836014
Median: mc = 105.4254819344

Inradius: r = 33.39443995264
Circumradius: R = 106.1054797524

Vertex coordinates: A[212.2021514757; 0] B[0; 0] C[34.31442615683; 77.20881054879]
Centroid: CG[82.17219254417; 25.73660351626]
Coordinates of the circumscribed circle: U[106.1010757378; -0.92659272821]
Coordinates of the inscribed circle: I[51.38657573784; 33.39443995264]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.5387791114° = 156°32'16″ = 0.40994927949 rad
∠ B' = β' = 113.9622208886° = 113°57'44″ = 1.15325768857 rad
∠ C' = γ' = 89.5° = 89°30' = 1.58795229731 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 = 84.49 ; ; b = 193.92 ; ; gamma = 90° 30' ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 84.49**2+193.92**2 - 2 * 84.49 * 193.92 * cos 90° 30' } ; ; c = 212.2 ; ;
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 = 84.49 ; ; b = 193.92 ; ; c = 212.2 ; ;

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

p = a+b+c = 84.49+193.92+212.2 = 490.61 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 490.61 }{ 2 } = 245.31 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 245.31 * (245.31-84.49)(245.31-193.92)(245.31-212.2) } ; ; T = sqrt{ 67106217.49 } = 8191.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8191.84 }{ 84.49 } = 193.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8191.84 }{ 193.92 } = 84.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8191.84 }{ 212.2 } = 77.21 ; ;

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{ 193.92**2+212.2**2-84.49**2 }{ 2 * 193.92 * 212.2 } ) = 23° 27'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 84.49**2+212.2**2-193.92**2 }{ 2 * 84.49 * 212.2 } ) = 66° 2'16" ; ; gamma = 180° - alpha - beta = 180° - 23° 27'44" - 66° 2'16" = 90° 30' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8191.84 }{ 245.31 } = 33.39 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 84.49 }{ 2 * sin 23° 27'44" } = 106.1 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 193.92**2+2 * 212.2**2 - 84.49**2 } }{ 2 } = 198.828 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 212.2**2+2 * 84.49**2 - 193.92**2 } }{ 2 } = 129.162 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 193.92**2+2 * 84.49**2 - 212.2**2 } }{ 2 } = 105.425 ; ;
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