Triangle calculator SSA

Please enter two sides and a non-included angle
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Obtuse isosceles triangle.

Sides: a = 90   b = 90   c = 163.1355401667

Area: T = 3102.487999463
Perimeter: p = 343.1355401667
Semiperimeter: s = 171.5687700833

Angle ∠ A = α = 25° = 0.4366332313 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 130° = 2.26989280276 rad

Height: ha = 68.94439998807
Height: hb = 68.94439998807
Height: hc = 38.03656435567

Median: ma = 123.8210756089
Median: mb = 123.8210756089
Median: mc = 38.03656435567

Inradius: r = 18.08331239188
Circumradius: R = 106.4799071242

Vertex coordinates: A[163.1355401667; 0] B[0; 0] C[81.56877008333; 38.03656435567]
Centroid: CG[81.56877008333; 12.67985478522]
Coordinates of the circumscribed circle: U[81.56877008333; -68.44334276852]
Coordinates of the inscribed circle: I[81.56877008333; 18.08331239188]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155° = 0.4366332313 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 50° = 2.26989280276 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 90 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 90**2 + c**2 -2 * 90 * c * cos (25° ) ; ; ; ; c**2 -163.135c =0 ; ; p=1; q=-163.135; r=0 ; ; D = q**2 - 4pr = 163.135**2 - 4 * 1 * 0 = 26613.1592769 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 163.14 ± sqrt{ 26613.16 } }{ 2 } ; ; c_{1,2} = 81.56770083 ± 81.5677008333 ; ; c_{1} = 163.135401663 ; ;
c_{2} = -3.29848148795E-9 ; ; ; ; text{ Factored form: } ; ; (c -163.135401663) (c +3.29848148795E-9) = 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 = 90 ; ; b = 90 ; ; c = 163.14 ; ;

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

p = a+b+c = 90+90+163.14 = 343.14 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 343.14 }{ 2 } = 171.57 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 171.57 * (171.57-90)(171.57-90)(171.57-163.14) } ; ; T = sqrt{ 9625382.12 } = 3102.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3102.48 }{ 90 } = 68.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3102.48 }{ 90 } = 68.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3102.48 }{ 163.14 } = 38.04 ; ;

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+163.14**2-90**2 }{ 2 * 90 * 163.14 } ) = 25° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90**2+163.14**2-90**2 }{ 2 * 90 * 163.14 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 25° - 25° = 130° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3102.48 }{ 171.57 } = 18.08 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 90 }{ 2 * sin 25° } = 106.48 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 163.14**2 - 90**2 } }{ 2 } = 123.821 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 163.14**2+2 * 90**2 - 90**2 } }{ 2 } = 123.821 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 90**2 - 163.14**2 } }{ 2 } = 38.036 ; ;
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