Triangle calculator SSA

Please enter two sides and a non-included angle
°


Right scalene triangle.

Sides: a = 49   b = 52   c = 17.40768951855

Area: T = 426.4698932046
Perimeter: p = 118.4076895186
Semiperimeter: s = 59.20334475928

Angle ∠ A = α = 70.44327862646° = 70°26'34″ = 1.22994585546 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 19.55772137354° = 19°33'26″ = 0.34113377722 rad

Height: ha = 17.40768951855
Height: hb = 16.40326512325
Height: hc = 49

Median: ma = 30.05441178543
Median: mb = 26
Median: mc = 49.76769569092

Inradius: r = 7.20334475928
Circumradius: R = 26

Vertex coordinates: A[17.40768951855; 0] B[0; 0] C[-0; 49]
Centroid: CG[5.80222983952; 16.33333333333]
Coordinates of the circumscribed circle: U[8.70334475928; 24.5]
Coordinates of the inscribed circle: I[7.20334475928; 7.20334475928]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 109.5577213735° = 109°33'26″ = 1.22994585546 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 160.4432786265° = 160°26'34″ = 0.34113377722 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 49 ; ; b = 52 ; ; beta = 90° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 52**2 = 49**2 + c**2 -2 * 52 * c * cos (90° ) ; ; ; ; c**2 -303 =0 ; ; p=1; q=-6.00076931582E-15; r=-303 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-303) = 1212 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 1212 } }{ 2 } = fraction{ ± 2 sqrt{ 303 } }{ 2 } ; ; c_{1,2} = 3.00038465791E-15 ± 17.4068951855 ; ;
c_{1} = sqrt{ 303} = 17.4068951855 ; ; c_{2} = - sqrt{ 303} = -17.4068951855 ; ; ; ; (c -17.4068951855) (c +17.4068951855) = 0 ; ; ; ; c>0 ; ;


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 = 49 ; ; b = 52 ; ; c = 17.41 ; ;

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

p = a+b+c = 49+52+17.41 = 118.41 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 118.41 }{ 2 } = 59.2 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 59.2 * (59.2-49)(59.2-52)(59.2-17.41) } ; ; T = sqrt{ 181875.75 } = 426.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 426.47 }{ 49 } = 17.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 426.47 }{ 52 } = 16.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 426.47 }{ 17.41 } = 49 ; ;

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{ 49**2-52**2-17.41**2 }{ 2 * 52 * 17.41 } ) = 70° 26'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 52**2-49**2-17.41**2 }{ 2 * 49 * 17.41 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17.41**2-49**2-52**2 }{ 2 * 52 * 49 } ) = 19° 33'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 426.47 }{ 59.2 } = 7.2 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 49 }{ 2 * sin 70° 26'34" } = 26 ; ;




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