Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 6.8   b = 5.9   c = 9.67548282401

Area: T = 19.81330281123
Perimeter: p = 22.37548282401
Semiperimeter: s = 11.187741412

Angle ∠ A = α = 43.96436248033° = 43°57'49″ = 0.76773100039 rad
Angle ∠ B = β = 37.03663751967° = 37°2'11″ = 0.64664066902 rad
Angle ∠ C = γ = 99° = 1.72878759595 rad

Height: ha = 5.82773612095
Height: hb = 6.7166280716
Height: hc = 4.09657891181

Median: ma = 7.25657667229
Median: mb = 7.8244234834
Median: mc = 4.13881668201

Inradius: r = 1.77110105213
Circumradius: R = 4.89877130955

Vertex coordinates: A[9.67548282401; 0] B[0; 0] C[5.42881222813; 4.09657891181]
Centroid: CG[5.03443168404; 1.36552630394]
Coordinates of the circumscribed circle: U[4.837741412; -0.7666171128]
Coordinates of the inscribed circle: I[5.287741412; 1.77110105213]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.0366375197° = 136°2'11″ = 0.76773100039 rad
∠ B' = β' = 142.9643624803° = 142°57'49″ = 0.64664066902 rad
∠ C' = γ' = 81° = 1.72878759595 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 = 6.8 ; ; b = 5.9 ; ; gamma = 99° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 6.8**2+5.9**2 - 2 * 6.8 * 5.9 * cos(99° ) } ; ; c = 9.67 ; ;


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 = 6.8 ; ; b = 5.9 ; ; c = 9.67 ; ;

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

p = a+b+c = 6.8+5.9+9.67 = 22.37 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.37 }{ 2 } = 11.19 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.19 * (11.19-6.8)(11.19-5.9)(11.19-9.67) } ; ; T = sqrt{ 392.56 } = 19.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19.81 }{ 6.8 } = 5.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19.81 }{ 5.9 } = 6.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19.81 }{ 9.67 } = 4.1 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 6.8**2-5.9**2-9.67**2 }{ 2 * 5.9 * 9.67 } ) = 43° 57'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.9**2-6.8**2-9.67**2 }{ 2 * 6.8 * 9.67 } ) = 37° 2'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.67**2-6.8**2-5.9**2 }{ 2 * 5.9 * 6.8 } ) = 99° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19.81 }{ 11.19 } = 1.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.8 }{ 2 * sin 43° 57'49" } = 4.9 ; ;




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