Triangle calculator SSA

Please enter two sides and a non-included angle
°


Right scalene triangle.

Sides: a = 68.685   b = 90.08444   c = 58.28986772741

Area: T = 2001.779889928
Perimeter: p = 217.0588077274
Semiperimeter: s = 108.5299038637

Angle ∠ A = α = 49.68108098205° = 49°40'51″ = 0.86770937064 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 40.31991901795° = 40°19'9″ = 0.70437026204 rad

Height: ha = 58.28986772741
Height: hb = 44.44222985397
Height: hc = 68.685

Median: ma = 67.65333606306
Median: mb = 45.04222
Median: mc = 74.61224768359

Inradius: r = 18.4454638637
Circumradius: R = 45.04222

Vertex coordinates: A[58.28986772741; 0] B[0; 0] C[-0; 68.685]
Centroid: CG[19.43295590914; 22.895]
Coordinates of the circumscribed circle: U[29.1444338637; 34.34325]
Coordinates of the inscribed circle: I[18.4454638637; 18.4454638637]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.3199190179° = 130°19'9″ = 0.86770937064 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 139.6810809821° = 139°40'51″ = 0.70437026204 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 68.69 ; ; b = 90.08 ; ; beta = 90° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90.08**2 = 68.69**2 + c**2 -2 * 90.08 * c * cos (90° ) ; ; ; ; c**2 -3397.57 =0 ; ; p=1; q=-8.41148653994E-15; r=-3397.56989836 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-3397.57) = 13590.2795934 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 13590.28 } }{ 2 } ; ; c_{1,2} = 4.20574326997E-15 ± 58.2886772741 ; ;
c_{1} = 58.2886772741 ; ; c_{2} = -58.2886772741 ; ; ; ; (c -58.2886772741) (c +58.2886772741) = 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 = 68.69 ; ; b = 90.08 ; ; c = 58.29 ; ;

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

p = a+b+c = 68.69+90.08+58.29 = 217.06 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 217.06 }{ 2 } = 108.53 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 108.53 * (108.53-68.69)(108.53-90.08)(108.53-58.29) } ; ; T = sqrt{ 4007118.76 } = 2001.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2001.78 }{ 68.69 } = 58.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2001.78 }{ 90.08 } = 44.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2001.78 }{ 58.29 } = 68.69 ; ;

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{ 68.69**2-90.08**2-58.29**2 }{ 2 * 90.08 * 58.29 } ) = 49° 40'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90.08**2-68.69**2-58.29**2 }{ 2 * 68.69 * 58.29 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 58.29**2-68.69**2-90.08**2 }{ 2 * 90.08 * 68.69 } ) = 40° 19'9" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2001.78 }{ 108.53 } = 18.44 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 68.69 }{ 2 * sin 49° 40'51" } = 45.04 ; ;




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