Triangle calculator SSA

Please enter two sides and a non-included angle
°


Obtuse scalene triangle.

Sides: a = 55   b = 90   c = 133.3277079713

Area: T = 1833.247734606
Perimeter: p = 278.3277079713
Semiperimeter: s = 139.1643539857

Angle ∠ A = α = 17.7921590573° = 17°47'30″ = 0.31105218347 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 132.2088409427° = 132°12'30″ = 2.30774720433 rad

Height: ha = 66.66435398566
Height: hb = 40.73988299124
Height: hc = 27.5

Median: ma = 110.3711214963
Median: mb = 91.51880588322
Median: mc = 33.44435711878

Inradius: r = 13.17333308016
Circumradius: R = 90

Vertex coordinates: A[133.3277079713; 0] B[0; 0] C[47.63113972081; 27.5]
Centroid: CG[60.31994923071; 9.16766666667]
Coordinates of the circumscribed circle: U[66.66435398566; -60.4654638044]
Coordinates of the inscribed circle: I[49.16435398566; 13.17333308016]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.2088409427° = 162°12'30″ = 0.31105218347 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 47.7921590573° = 47°47'30″ = 2.30774720433 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 55 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 55**2 + c**2 -2 * 55 * c * cos (30° ) ; ; ; ; c**2 -95.263c -5075 =0 ; ; p=1; q=-95.263; r=-5075 ; ; D = q**2 - 4pr = 95.263**2 - 4 * 1 * (-5075) = 29375 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 95.26 ± sqrt{ 29375 } }{ 2 } = fraction{ 95.26 ± 25 sqrt{ 47 } }{ 2 } ; ; c_{1,2} = 47.63139721 ± 85.695682505 ; ;
c_{1} = 133.327079715 ; ; c_{2} = -38.064285295 ; ; ; ; text{ Factored form: } ; ; (c -133.327079715) (c +38.064285295) = 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 = 55 ; ; b = 90 ; ; c = 133.33 ; ;

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

p = a+b+c = 55+90+133.33 = 278.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 278.33 }{ 2 } = 139.16 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 139.16 * (139.16-55)(139.16-90)(139.16-133.33) } ; ; T = sqrt{ 3360795.83 } = 1833.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1833.25 }{ 55 } = 66.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1833.25 }{ 90 } = 40.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1833.25 }{ 133.33 } = 27.5 ; ;

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{ 90**2+133.33**2-55**2 }{ 2 * 90 * 133.33 } ) = 17° 47'30" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 55**2+133.33**2-90**2 }{ 2 * 55 * 133.33 } ) = 30° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 55**2+90**2-133.33**2 }{ 2 * 55 * 90 } ) = 132° 12'30" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1833.25 }{ 139.16 } = 13.17 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 55 }{ 2 * sin 17° 47'30" } = 90 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 133.33**2 - 55**2 } }{ 2 } = 110.371 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 133.33**2+2 * 55**2 - 90**2 } }{ 2 } = 91.518 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 55**2 - 133.33**2 } }{ 2 } = 33.444 ; ;
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