Triangle calculator SSA

Please enter two sides and a non-included angle
°


Obtuse scalene triangle.

Sides: a = 50   b = 60   c = 88.96883866258

Area: T = 1429.694441442
Perimeter: p = 198.9688386626
Semiperimeter: s = 99.48441933129

Angle ∠ A = α = 32.38884338246° = 32°23'18″ = 0.56552848098 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 107.6121566175° = 107°36'42″ = 1.8788176143 rad

Height: ha = 57.18877765769
Height: hb = 47.65664804807
Height: hc = 32.13993804843

Median: ma = 71.64327729042
Median: mb = 65.63329712065
Median: mc = 32.72985280039

Inradius: r = 14.37110710899
Circumradius: R = 46.67217148058

Vertex coordinates: A[88.96883866258; 0] B[0; 0] C[38.30222221559; 32.13993804843]
Centroid: CG[42.42435362606; 10.71331268281]
Coordinates of the circumscribed circle: U[44.48441933129; -14.12111015228]
Coordinates of the inscribed circle: I[39.48441933129; 14.37110710899]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.6121566175° = 147°36'42″ = 0.56552848098 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 72.38884338246° = 72°23'18″ = 1.8788176143 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 50 ; ; b = 60 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 60**2 = 50**2 + c**2 -2 * 60 * c * cos (40° ) ; ; ; ; c**2 -76.604c -1100 =0 ; ; p=1; q=-76.6044443119; r=-1100 ; ; D = q**2 - 4pr = 76.604**2 - 4 * 1 * (-1100) = 10268.2408883 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 76.6 ± sqrt{ 10268.24 } }{ 2 } ; ; c_{1,2} = 38.3022221559 ± 50.6661644698 ; ; c_{1} = 88.9683866258 ; ;
c_{2} = -12.3639423139 ; ; ; ; (c -88.9683866258) (c +12.3639423139) = 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 = 50 ; ; b = 60 ; ; c = 88.97 ; ;

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

p = a+b+c = 50+60+88.97 = 198.97 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 198.97 }{ 2 } = 99.48 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 99.48 * (99.48-50)(99.48-60)(99.48-88.97) } ; ; T = sqrt{ 2044026.12 } = 1429.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1429.69 }{ 50 } = 57.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1429.69 }{ 60 } = 47.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1429.69 }{ 88.97 } = 32.14 ; ;

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{ 50**2-60**2-88.97**2 }{ 2 * 60 * 88.97 } ) = 32° 23'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-50**2-88.97**2 }{ 2 * 50 * 88.97 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 88.97**2-50**2-60**2 }{ 2 * 60 * 50 } ) = 107° 36'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1429.69 }{ 99.48 } = 14.37 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 32° 23'18" } = 46.67 ; ;




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