Triangle calculator SSA

Please enter two sides and a non-included angle
°


Acute scalene triangle.

Sides: a = 85   b = 90   c = 94.2810787943

Area: T = 3470.106619152
Perimeter: p = 269.2810787943
Semiperimeter: s = 134.6440393972

Angle ∠ A = α = 54.87664076349° = 54°52'35″ = 0.95877739949 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 65.12435923651° = 65°7'25″ = 1.13766211075 rad

Height: ha = 81.65495574475
Height: hb = 77.11334709226
Height: hc = 73.61221593217

Median: ma = 81.78113150272
Median: mb = 77.66655231591
Median: mc = 73.7588275849

Inradius: r = 25.77331434762
Circumradius: R = 51.96215242271

Vertex coordinates: A[94.2810787943; 0] B[0; 0] C[42.5; 73.61221593217]
Centroid: CG[45.5943595981; 24.53773864406]
Coordinates of the circumscribed circle: U[47.14403939715; 21.8588253732]
Coordinates of the inscribed circle: I[44.64403939715; 25.77331434762]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.1243592365° = 125°7'25″ = 0.95877739949 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 114.8766407635° = 114°52'35″ = 1.13766211075 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 85 ; ; b = 90 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 85**2 + c**2 -2 * 90 * c * cos (60° ) ; ; ; ; c**2 -85c -875 =0 ; ; p=1; q=-85; r=-875 ; ; D = q**2 - 4pr = 85**2 - 4 * 1 * (-875) = 10725 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 85 ± sqrt{ 10725 } }{ 2 } ; ; c_{1,2} = 42.5 ± 51.780787943 ; ; c_{1} = 94.280787943 ; ; c_{2} = -9.28078794302 ; ;
 ; ; (c -94.280787943) (c +9.28078794302) = 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 = 85 ; ; b = 90 ; ; c = 94.28 ; ;

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

p = a+b+c = 85+90+94.28 = 269.28 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 269.28 }{ 2 } = 134.64 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 134.64 * (134.64-85)(134.64-90)(134.64-94.28) } ; ; T = sqrt{ 12041636.98 } = 3470.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3470.11 }{ 85 } = 81.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3470.11 }{ 90 } = 77.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3470.11 }{ 94.28 } = 73.61 ; ;

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{ 85**2-90**2-94.28**2 }{ 2 * 90 * 94.28 } ) = 54° 52'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-85**2-94.28**2 }{ 2 * 85 * 94.28 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 94.28**2-85**2-90**2 }{ 2 * 90 * 85 } ) = 65° 7'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3470.11 }{ 134.64 } = 25.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 85 }{ 2 * sin 54° 52'35" } = 51.96 ; ;




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