Triangle calculator SSA

Please enter two sides and a non-included angle
°


Right scalene triangle.

Sides: a = 160   b = 192   c = 106.1321993291

Area: T = 8490.559946331
Perimeter: p = 458.1321993291
Semiperimeter: s = 229.0665996646

Angle ∠ A = α = 56.44326902381° = 56°26'34″ = 0.98551107833 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 33.55773097619° = 33°33'26″ = 0.58656855435 rad

Height: ha = 106.1321993291
Height: hb = 88.44333277428
Height: hc = 160

Median: ma = 132.9065981807
Median: mb = 96
Median: mc = 168.5770460046

Inradius: r = 37.06659966457
Circumradius: R = 96

Vertex coordinates: A[106.1321993291; 0] B[0; 0] C[-0; 160]
Centroid: CG[35.37773310971; 53.33333333333]
Coordinates of the circumscribed circle: U[53.06659966457; 80]
Coordinates of the inscribed circle: I[37.06659966457; 37.06659966457]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.5577309762° = 123°33'26″ = 0.98551107833 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 146.4432690238° = 146°26'34″ = 0.58656855435 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 160 ; ; b = 192 ; ; beta = 90° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 192**2 = 160**2 + c**2 -2 * 192 * c * cos (90° ) ; ; ; ; c**2 -11264 =0 ; ; p=1; q=-1.95943487864E-14; r=-11264 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-11264) = 45056 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 45056 } }{ 2 } = fraction{ ± 64 sqrt{ 11 } }{ 2 } ; ; c_{1,2} = 9.79717439318E-15 ± 106.131993291 ; ;
c_{1} = 32 sqrt{ 11} = 106.131993291 ; ; c_{2} = - 32 sqrt{ 11} = -106.131993291 ; ; ; ; (c -106.131993291) (c +106.131993291) = 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 = 160 ; ; b = 192 ; ; c = 106.13 ; ;

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

p = a+b+c = 160+192+106.13 = 458.13 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 458.13 }{ 2 } = 229.07 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 229.07 * (229.07-160)(229.07-192)(229.07-106.13) } ; ; T = sqrt{ 72089600 } = 8490.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8490.56 }{ 160 } = 106.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8490.56 }{ 192 } = 88.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8490.56 }{ 106.13 } = 160 ; ;

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{ 160**2-192**2-106.13**2 }{ 2 * 192 * 106.13 } ) = 56° 26'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 192**2-160**2-106.13**2 }{ 2 * 160 * 106.13 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 106.13**2-160**2-192**2 }{ 2 * 192 * 160 } ) = 33° 33'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8490.56 }{ 229.07 } = 37.07 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 160 }{ 2 * sin 56° 26'34" } = 96 ; ;




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