Triangle calculator SAS

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


Right scalene triangle.

Sides: a = 38   b = 120   c = 125.8732951821

Area: T = 2280
Perimeter: p = 283.8732951821
Semiperimeter: s = 141.936647591

Angle ∠ A = α = 17.57112587783° = 17°34'17″ = 0.30766763194 rad
Angle ∠ B = β = 72.42987412217° = 72°25'43″ = 1.26441200074 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 120
Height: hb = 38
Height: hc = 36.22770045633

Median: ma = 121.4954855858
Median: mb = 71.02111236183
Median: mc = 62.93664759102

Inradius: r = 16.06435240898
Circumradius: R = 62.93664759102

Vertex coordinates: A[125.8732951821; 0] B[0; 0] C[11.47218847784; 36.22770045633]
Centroid: CG[45.78216121996; 12.07656681878]
Coordinates of the circumscribed circle: U[62.93664759102; 0]
Coordinates of the inscribed circle: I[21.93664759102; 16.06435240898]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.4298741222° = 162°25'43″ = 0.30766763194 rad
∠ B' = β' = 107.5711258778° = 107°34'17″ = 1.26441200074 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 38 ; ; b = 120 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 38**2+120**2 - 2 * 38 * 120 * cos 90° } ; ; c = 125.87 ; ;


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 = 38 ; ; b = 120 ; ; c = 125.87 ; ;

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

p = a+b+c = 38+120+125.87 = 283.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 283.87 }{ 2 } = 141.94 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 141.94 * (141.94-38)(141.94-120)(141.94-125.87) } ; ; T = sqrt{ 5198400 } = 2280 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2280 }{ 38 } = 120 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2280 }{ 120 } = 38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2280 }{ 125.87 } = 36.23 ; ;

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{ 120**2+125.87**2-38**2 }{ 2 * 120 * 125.87 } ) = 17° 34'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38**2+125.87**2-120**2 }{ 2 * 38 * 125.87 } ) = 72° 25'43" ; ; gamma = 180° - alpha - beta = 180° - 17° 34'17" - 72° 25'43" = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2280 }{ 141.94 } = 16.06 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 38 }{ 2 * sin 17° 34'17" } = 62.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 120**2+2 * 125.87**2 - 38**2 } }{ 2 } = 121.495 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 125.87**2+2 * 38**2 - 120**2 } }{ 2 } = 71.021 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 120**2+2 * 38**2 - 125.87**2 } }{ 2 } = 62.936 ; ;
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