Triangle calculator SSA

Please enter two sides and a non-included angle
°


Acute scalene triangle.

Sides: a = 3.5   b = 4.2   c = 4.6577318352

Area: T = 7.05883730111
Perimeter: p = 12.3577318352
Semiperimeter: s = 6.1798659176

Angle ∠ A = α = 46.19440077316° = 46°11'38″ = 0.80662375296 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 73.80659922684° = 73°48'22″ = 1.28881575728 rad

Height: ha = 4.03333560064
Height: hb = 3.36111300053
Height: hc = 3.03110889132

Median: ma = 4.07546542327
Median: mb = 3.5444052358
Median: mc = 3.08658299438

Inradius: r = 1.14223794079
Circumradius: R = 2.42548711306

Vertex coordinates: A[4.6577318352; 0] B[0; 0] C[1.75; 3.03110889132]
Centroid: CG[2.1365772784; 1.01103629711]
Coordinates of the circumscribed circle: U[2.3298659176; 0.676627394]
Coordinates of the inscribed circle: I[1.9798659176; 1.14223794079]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.8065992268° = 133°48'22″ = 0.80662375296 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 106.1944007732° = 106°11'38″ = 1.28881575728 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 3.5 ; ; b = 4.2 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 4.2**2 = 3.5**2 + c**2 -2 * 4.2 * c * cos (60° ) ; ; ; ; c**2 -3.5c -5.39 =0 ; ; p=1; q=-3.5; r=-5.39 ; ; D = q**2 - 4pr = 3.5**2 - 4 * 1 * (-5.39) = 33.81 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 3.5 ± sqrt{ 33.81 } }{ 2 } ; ; c_{1,2} = 1.75 ± 2.90731835202 ; ; c_{1} = 4.65731835202 ; ; c_{2} = -1.15731835202 ; ;
 ; ; (c -4.65731835202) (c +1.15731835202) = 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 = 3.5 ; ; b = 4.2 ; ; c = 4.66 ; ;

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

p = a+b+c = 3.5+4.2+4.66 = 12.36 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.36 }{ 2 } = 6.18 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.18 * (6.18-3.5)(6.18-4.2)(6.18-4.66) } ; ; T = sqrt{ 49.82 } = 7.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.06 }{ 3.5 } = 4.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.06 }{ 4.2 } = 3.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.06 }{ 4.66 } = 3.03 ; ;

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{ 3.5**2-4.2**2-4.66**2 }{ 2 * 4.2 * 4.66 } ) = 46° 11'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.2**2-3.5**2-4.66**2 }{ 2 * 3.5 * 4.66 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.66**2-3.5**2-4.2**2 }{ 2 * 4.2 * 3.5 } ) = 73° 48'22" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.06 }{ 6.18 } = 1.14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.5 }{ 2 * sin 46° 11'38" } = 2.42 ; ;




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