Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 4.7   b = 6.1   c = 4.49437972845

Area: T = 10.48439553127
Perimeter: p = 15.29437972845
Semiperimeter: s = 7.64768986422

Angle ∠ A = α = 49.89992217605° = 49°53'57″ = 0.87109057139 rad
Angle ∠ B = β = 83.10107782395° = 83°6'3″ = 1.45503821912 rad
Angle ∠ C = γ = 47° = 0.82203047484 rad

Height: ha = 4.46112575799
Height: hb = 3.43773623976
Height: hc = 4.66659671761

Median: ma = 4.81545204348
Median: mb = 3.44108730022
Median: mc = 4.96599845253

Inradius: r = 1.37110074899
Circumradius: R = 3.07222462158

Vertex coordinates: A[4.49437972845; 0] B[0; 0] C[0.56545797655; 4.66659671761]
Centroid: CG[1.68661256833; 1.5555322392]
Coordinates of the circumscribed circle: U[2.24768986422; 2.09552668809]
Coordinates of the inscribed circle: I[1.54768986422; 1.37110074899]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.1010778239° = 130°6'3″ = 0.87109057139 rad
∠ B' = β' = 96.89992217605° = 96°53'57″ = 1.45503821912 rad
∠ C' = γ' = 133° = 0.82203047484 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 = 4.7 ; ; b = 6.1 ; ; gamma = 47° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 4.7**2+6.1**2 - 2 * 4.7 * 6.1 * cos 47° } ; ; c = 4.49 ; ;


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 = 4.7 ; ; b = 6.1 ; ; c = 4.49 ; ;

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

p = a+b+c = 4.7+6.1+4.49 = 15.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.29 }{ 2 } = 7.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.65 * (7.65-4.7)(7.65-6.1)(7.65-4.49) } ; ; T = sqrt{ 109.91 } = 10.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10.48 }{ 4.7 } = 4.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10.48 }{ 6.1 } = 3.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10.48 }{ 4.49 } = 4.67 ; ;

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{ 6.1**2+4.49**2-4.7**2 }{ 2 * 6.1 * 4.49 } ) = 49° 53'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.7**2+4.49**2-6.1**2 }{ 2 * 4.7 * 4.49 } ) = 83° 6'3" ; ; gamma = 180° - alpha - beta = 180° - 49° 53'57" - 83° 6'3" = 47° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10.48 }{ 7.65 } = 1.37 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.7 }{ 2 * sin 49° 53'57" } = 3.07 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.1**2+2 * 4.49**2 - 4.7**2 } }{ 2 } = 4.815 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.49**2+2 * 4.7**2 - 6.1**2 } }{ 2 } = 3.441 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.1**2+2 * 4.7**2 - 4.49**2 } }{ 2 } = 4.96 ; ;
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