Triangle calculator SSA

Please enter two sides and a non-included angle
°


Obtuse scalene triangle.

Sides: a = 40.7   b = 52.5   c = 84.485503136

Area: T = 823.0010736419
Perimeter: p = 177.685503136
Semiperimeter: s = 88.843251568

Angle ∠ A = α = 21.78334832618° = 21°47'1″ = 0.38801935055 rad
Angle ∠ B = β = 28.6° = 28°36' = 0.49991641661 rad
Angle ∠ C = γ = 129.6176516738° = 129°36'59″ = 2.2622234982 rad

Height: ha = 40.44222966299
Height: hb = 31.35224090064
Height: hc = 19.48327586182

Median: ma = 67.32765383185
Median: mb = 60.89437005112
Median: mc = 20.5411174967

Inradius: r = 9.26435910872
Circumradius: R = 54.83769469097

Vertex coordinates: A[84.485503136; 0] B[0; 0] C[35.73439070999; 19.48327586182]
Centroid: CG[40.07329794867; 6.49442528727]
Coordinates of the circumscribed circle: U[42.243251568; -34.96765642493]
Coordinates of the inscribed circle: I[36.343251568; 9.26435910872]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.2176516738° = 158°12'59″ = 0.38801935055 rad
∠ B' = β' = 151.4° = 151°24' = 0.49991641661 rad
∠ C' = γ' = 50.38334832618° = 50°23'1″ = 2.2622234982 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 40.7 ; ; b = 52.5 ; ; beta = 28° 36' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 52.5**2 = 40.7**2 + c**2 -2 * 52.5 * c * cos (28° 36') ; ; ; ; c**2 -71.468c -1099.76 =0 ; ; p=1; q=-71.4678141998; r=-1099.76 ; ; D = q**2 - 4pr = 71.468**2 - 4 * 1 * (-1099.76) = 9506.6884665 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 71.47 ± sqrt{ 9506.69 } }{ 2 } ; ; c_{1,2} = 35.7339070999 ± 48.7511242601 ; ;
c_{1} = 84.48503136 ; ; c_{2} = -13.0172171602 ; ; ; ; (c -84.48503136) (c +13.0172171602) = 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 = 40.7 ; ; b = 52.5 ; ; c = 84.49 ; ;

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

p = a+b+c = 40.7+52.5+84.49 = 177.69 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 177.69 }{ 2 } = 88.84 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 88.84 * (88.84-40.7)(88.84-52.5)(88.84-84.49) } ; ; T = sqrt{ 677330.21 } = 823 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 823 }{ 40.7 } = 40.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 823 }{ 52.5 } = 31.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 823 }{ 84.49 } = 19.48 ; ;

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{ 40.7**2-52.5**2-84.49**2 }{ 2 * 52.5 * 84.49 } ) = 21° 47'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 52.5**2-40.7**2-84.49**2 }{ 2 * 40.7 * 84.49 } ) = 28° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 84.49**2-40.7**2-52.5**2 }{ 2 * 52.5 * 40.7 } ) = 129° 36'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 823 }{ 88.84 } = 9.26 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 40.7 }{ 2 * sin 21° 47'1" } = 54.84 ; ;




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