Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 51   b = 110   c = 148.0310662299

Area: T = 2148.755466295
Perimeter: p = 309.0310662299
Semiperimeter: s = 154.515533115

Angle ∠ A = α = 15.30327946029° = 15°18'10″ = 0.26770841506 rad
Angle ∠ B = β = 34.69772053971° = 34°41'50″ = 0.60655804754 rad
Angle ∠ C = γ = 130° = 2.26989280276 rad

Height: ha = 84.26548887431
Height: hb = 39.06882665991
Height: hc = 29.03112105557

Median: ma = 127.8921706105
Median: mb = 96.08334974922
Median: mc = 43.26992818848

Inradius: r = 13.90664172271
Circumradius: R = 96.62201528049

Vertex coordinates: A[148.0310662299; 0] B[0; 0] C[41.93107621404; 29.03112105557]
Centroid: CG[63.32204748131; 9.67770701852]
Coordinates of the circumscribed circle: U[74.01553311495; -62.1066237069]
Coordinates of the inscribed circle: I[44.51553311495; 13.90664172271]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.6977205397° = 164°41'50″ = 0.26770841506 rad
∠ B' = β' = 145.3032794603° = 145°18'10″ = 0.60655804754 rad
∠ C' = γ' = 50° = 2.26989280276 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 = 51 ; ; b = 110 ; ; gamma = 130° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 51**2+110**2 - 2 * 51 * 110 * cos(130° ) } ; ; c = 148.03 ; ;


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 = 51 ; ; b = 110 ; ; c = 148.03 ; ;

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

p = a+b+c = 51+110+148.03 = 309.03 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 309.03 }{ 2 } = 154.52 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 154.52 * (154.52-51)(154.52-110)(154.52-148.03) } ; ; T = sqrt{ 4617146.6 } = 2148.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2148.75 }{ 51 } = 84.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2148.75 }{ 110 } = 39.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2148.75 }{ 148.03 } = 29.03 ; ;

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{ 51**2-110**2-148.03**2 }{ 2 * 110 * 148.03 } ) = 15° 18'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 110**2-51**2-148.03**2 }{ 2 * 51 * 148.03 } ) = 34° 41'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 148.03**2-51**2-110**2 }{ 2 * 110 * 51 } ) = 130° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2148.75 }{ 154.52 } = 13.91 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 51 }{ 2 * sin 15° 18'10" } = 96.62 ; ;




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