Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 20.4   b = 17.7   c = 29.02444847202

Area: T = 178.3177253011
Perimeter: p = 67.12444847202
Semiperimeter: s = 33.56222423601

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 = 17.48220836285
Height: hb = 20.14988421481
Height: hc = 12.28773673542

Median: ma = 21.76773001688
Median: mb = 23.4732704502
Median: mc = 12.41545004604

Inradius: r = 5.31330315638
Circumradius: R = 14.69331392866

Vertex coordinates: A[29.02444847202; 0] B[0; 0] C[16.28443668438; 12.28773673542]
Centroid: CG[15.10329505213; 4.09657891181]
Coordinates of the circumscribed circle: U[14.51222423601; -2.29985133841]
Coordinates of the inscribed circle: I[15.86222423601; 5.31330315638]

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 = 20.4 ; ; b = 17.7 ; ; gamma = 99° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 20.4**2+17.7**2 - 2 * 20.4 * 17.7 * cos 99° } ; ; c = 29.02 ; ;


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 = 20.4 ; ; b = 17.7 ; ; c = 29.02 ; ;

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

p = a+b+c = 20.4+17.7+29.02 = 67.12 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67.12 }{ 2 } = 33.56 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.56 * (33.56-20.4)(33.56-17.7)(33.56-29.02) } ; ; T = sqrt{ 31797.04 } = 178.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 178.32 }{ 20.4 } = 17.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 178.32 }{ 17.7 } = 20.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 178.32 }{ 29.02 } = 12.29 ; ;

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{ 17.7**2+29.02**2-20.4**2 }{ 2 * 17.7 * 29.02 } ) = 43° 57'49" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20.4**2+29.02**2-17.7**2 }{ 2 * 20.4 * 29.02 } ) = 37° 2'11" ; ; gamma = 180° - alpha - beta = 180° - 43° 57'49" - 37° 2'11" = 99° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 178.32 }{ 33.56 } = 5.31 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20.4 }{ 2 * sin 43° 57'49" } = 14.69 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.7**2+2 * 29.02**2 - 20.4**2 } }{ 2 } = 21.767 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.02**2+2 * 20.4**2 - 17.7**2 } }{ 2 } = 23.473 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.7**2+2 * 20.4**2 - 29.02**2 } }{ 2 } = 12.415 ; ;
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