Triangle calculator SSA

Please enter two sides and a non-included angle
°


Obtuse scalene triangle.

Sides: a = 3.2   b = 8.4   c = 10.09112791512

Area: T = 12.36985893084
Perimeter: p = 21.69112791512
Semiperimeter: s = 10.84656395756

Angle ∠ A = α = 16.96773348748° = 16°58'2″ = 0.29661358589 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 113.0332665125° = 113°1'58″ = 1.97327921687 rad

Height: ha = 7.73303683177
Height: hb = 2.94549022163
Height: hc = 2.4511342218

Median: ma = 9.14553243493
Median: mb = 6.19765278547
Median: mc = 3.86554264025

Inradius: r = 1.14404204632
Circumradius: R = 5.48327106152

Vertex coordinates: A[10.09112791512; 0] B[0; 0] C[2.0576920351; 2.4511342218]
Centroid: CG[4.04993998341; 0.81771140727]
Coordinates of the circumscribed circle: U[5.04656395756; -2.1455142644]
Coordinates of the inscribed circle: I[2.44656395756; 1.14404204632]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.0332665125° = 163°1'58″ = 0.29661358589 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 66.96773348748° = 66°58'2″ = 1.97327921687 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 3.2 ; ; b = 8.4 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 8.4**2 = 3.2**2 + c**2 -2 * 8.4 * c * cos (50° ) ; ; ; ; c**2 -4.114c -60.32 =0 ; ; p=1; q=-4.11384070199; r=-60.32 ; ; D = q**2 - 4pr = 4.114**2 - 4 * 1 * (-60.32) = 258.203685321 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.11 ± sqrt{ 258.2 } }{ 2 } ; ; c_{1,2} = 2.056920351 ± 8.0343588002 ; ; c_{1} = 10.0912791512 ; ;
c_{2} = -5.9774384492 ; ; ; ; (c -10.0912791512) (c +5.9774384492) = 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 = 3.2 ; ; b = 8.4 ; ; c = 10.09 ; ;

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

p = a+b+c = 3.2+8.4+10.09 = 21.69 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.69 }{ 2 } = 10.85 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.85 * (10.85-3.2)(10.85-8.4)(10.85-10.09) } ; ; T = sqrt{ 152.98 } = 12.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12.37 }{ 3.2 } = 7.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12.37 }{ 8.4 } = 2.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12.37 }{ 10.09 } = 2.45 ; ;

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{ 3.2**2-8.4**2-10.09**2 }{ 2 * 8.4 * 10.09 } ) = 16° 58'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.4**2-3.2**2-10.09**2 }{ 2 * 3.2 * 10.09 } ) = 50° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.09**2-3.2**2-8.4**2 }{ 2 * 8.4 * 3.2 } ) = 113° 1'58" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12.37 }{ 10.85 } = 1.14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.2 }{ 2 * sin 16° 58'2" } = 5.48 ; ;




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