Triangle calculator SSA

Please enter two sides and a non-included angle
°


Right scalene triangle.

Sides: a = 3.5   b = 4.2   c = 2.32216373532

Area: T = 4.06328653682
Perimeter: p = 10.02216373532
Semiperimeter: s = 5.01108186766

Angle ∠ A = α = 56.44326902381° = 56°26'34″ = 0.98551107833 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 33.55773097619° = 33°33'26″ = 0.58656855435 rad

Height: ha = 2.32216373532
Height: hb = 1.93546977944
Height: hc = 3.5

Median: ma = 2.9077318352
Median: mb = 2.1
Median: mc = 3.68774788135

Inradius: r = 0.81108186766
Circumradius: R = 2.1

Vertex coordinates: A[2.32216373532; 0] B[0; 0] C[-0; 3.5]
Centroid: CG[0.77438791177; 1.16766666667]
Coordinates of the circumscribed circle: U[1.16108186766; 1.75]
Coordinates of the inscribed circle: I[0.81108186766; 0.81108186766]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.5577309762° = 123°33'26″ = 0.98551107833 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 146.4432690238° = 146°26'34″ = 0.58656855435 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 3.5 ; ; b = 4.2 ; ; beta = 90° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 4.2**2 = 3.5**2 + c**2 -2 * 4.2 * c * cos (90° ) ; ; ; ; c**2 -5.39 =0 ; ; p=1; q=-4.28626379702E-16; r=-5.39 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-5.39) = 21.56 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 21.56 } }{ 2 } ; ; c_{1,2} = 2.14313189851E-16 ± 2.32163735325 ; ; c_{1} = 2.32163735325 ; ;
c_{2} = -2.32163735325 ; ; ; ; (c -2.32163735325) (c +2.32163735325) = 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 = 2.32 ; ;

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

p = a+b+c = 3.5+4.2+2.32 = 10.02 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.02 }{ 2 } = 5.01 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.01 * (5.01-3.5)(5.01-4.2)(5.01-2.32) } ; ; T = sqrt{ 16.51 } = 4.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.06 }{ 3.5 } = 2.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.06 }{ 4.2 } = 1.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.06 }{ 2.32 } = 3.5 ; ;

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**2-2.32**2 }{ 2 * 4.2 * 2.32 } ) = 56° 26'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.2**2-3.5**2-2.32**2 }{ 2 * 3.5 * 2.32 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.32**2-3.5**2-4.2**2 }{ 2 * 4.2 * 3.5 } ) = 33° 33'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.06 }{ 5.01 } = 0.81 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.5 }{ 2 * sin 56° 26'34" } = 2.1 ; ;




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