Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 40   b = 49   c = 40.52201815056

Area: T = 782.6632799846
Perimeter: p = 129.5220181506
Semiperimeter: s = 64.76600907528

Angle ∠ A = α = 52.03546518095° = 52°2'5″ = 0.90881759992 rad
Angle ∠ B = β = 74.96553481905° = 74°57'55″ = 1.30883921508 rad
Angle ∠ C = γ = 53° = 0.92550245036 rad

Height: ha = 39.13331399923
Height: hb = 31.94554204019
Height: hc = 38.6310764758

Median: ma = 40.26771398863
Median: mb = 31.94882793687
Median: mc = 39.8755164234

Inradius: r = 12.08655729315
Circumradius: R = 25.3688382069

Vertex coordinates: A[40.52201815056; 0] B[0; 0] C[10.37661271297; 38.6310764758]
Centroid: CG[16.96554362118; 12.8776921586]
Coordinates of the circumscribed circle: U[20.26600907528; 15.26770734422]
Coordinates of the inscribed circle: I[15.76600907528; 12.08655729315]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.965534819° = 127°57'55″ = 0.90881759992 rad
∠ B' = β' = 105.035465181° = 105°2'5″ = 1.30883921508 rad
∠ C' = γ' = 127° = 0.92550245036 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 = 40 ; ; b = 49 ; ; gamma = 53° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 40**2+49**2 - 2 * 40 * 49 * cos 53° } ; ; c = 40.52 ; ;
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 = 49 ; ; c = 40.52 ; ;

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

p = a+b+c = 40+49+40.52 = 129.52 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 129.52 }{ 2 } = 64.76 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 64.76 * (64.76-40)(64.76-49)(64.76-40.52) } ; ; T = sqrt{ 612561.06 } = 782.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 782.66 }{ 40 } = 39.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 782.66 }{ 49 } = 31.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 782.66 }{ 40.52 } = 38.63 ; ;

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{ 49**2+40.52**2-40**2 }{ 2 * 49 * 40.52 } ) = 52° 2'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40**2+40.52**2-49**2 }{ 2 * 40 * 40.52 } ) = 74° 57'55" ; ;
 gamma = 180° - alpha - beta = 180° - 52° 2'5" - 74° 57'55" = 53° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 782.66 }{ 64.76 } = 12.09 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 40 }{ 2 * sin 52° 2'5" } = 25.37 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 40.52**2 - 40**2 } }{ 2 } = 40.267 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 40.52**2+2 * 40**2 - 49**2 } }{ 2 } = 31.948 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 40**2 - 40.52**2 } }{ 2 } = 39.875 ; ;
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