Triangle calculator SSA

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Triangle has two solutions with side c=26.9065779993 and with side c=9.70105178838

#1 Acute scalene triangle.

Sides: a = 45   b = 42   c = 26.9065779993

Area: T = 553.04221946814
Perimeter: p = 113.9065779993
Semiperimeter: s = 56.95328899965

Angle ∠ A = α = 78.18107714952° = 78°10'51″ = 1.36545118743 rad
Angle ∠ B = β = 66° = 1.15219173063 rad
Angle ∠ C = γ = 35.81992285048° = 35°49'9″ = 0.62551634729 rad

Height: ha = 24.5879653097
Height: hb = 26.33553426039
Height: hc = 41.11095455939

Median: ma = 27.16108265433
Median: mb = 30.5532585791
Median: mc = 41.39546826385

Inradius: r = 9.7110520304
Circumradius: R = 22.98773618486

Vertex coordinates: A[26.9065779993; 0] B[0; 0] C[18.30331489384; 41.11095455939]
Centroid: CG[15.07696429771; 13.70331818646]
Coordinates of the circumscribed circle: U[13.45328899965; 18.64397037396]
Coordinates of the inscribed circle: I[14.95328899965; 9.7110520304]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 101.81992285048° = 101°49'9″ = 1.36545118743 rad
∠ B' = β' = 114° = 1.15219173063 rad
∠ C' = γ' = 144.18107714952° = 144°10'51″ = 0.62551634729 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 45 ; ; b = 42 ; ; beta = 66° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 45**2 + c**2 -2 * 45 * c * cos (66° ) ; ; ; ; c**2 -36.606c +261 =0 ; ; p=1; q=-36.606; r=261 ; ; D = q**2 - 4pr = 36.606**2 - 4 * 1 * 261 = 296.02104424663 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 36.61 ± sqrt{ 296.02 } }{ 2 } ; ; c_{1,2} = 18.30314894 ± 8.6026310546051 ; ; c_{1} = 26.905779994605 ; ;
c_{2} = 9.7005178853949 ; ; ; ; text{ Factored form: } ; ; (c -26.905779994605) (c -9.7005178853949) = 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 = 45 ; ; b = 42 ; ; c = 26.91 ; ;

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

p = a+b+c = 45+42+26.91 = 113.91 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 113.91 }{ 2 } = 56.95 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 56.95 * (56.95-45)(56.95-42)(56.95-26.91) } ; ; T = sqrt{ 305855.67 } = 553.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 553.04 }{ 45 } = 24.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 553.04 }{ 42 } = 26.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 553.04 }{ 26.91 } = 41.11 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 42**2+26.91**2-45**2 }{ 2 * 42 * 26.91 } ) = 78° 10'51" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+26.91**2-42**2 }{ 2 * 45 * 26.91 } ) = 66° ; ; gamma = 180° - alpha - beta = 180° - 78° 10'51" - 66° = 35° 49'9" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 553.04 }{ 56.95 } = 9.71 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45 }{ 2 * sin 78° 10'51" } = 22.99 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 26.91**2 - 45**2 } }{ 2 } = 27.161 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.91**2+2 * 45**2 - 42**2 } }{ 2 } = 30.553 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 45**2 - 26.91**2 } }{ 2 } = 41.395 ; ;







#2 Obtuse scalene triangle.

Sides: a = 45   b = 42   c = 9.70105178838

Area: T = 199.39219411145
Perimeter: p = 96.70105178838
Semiperimeter: s = 48.35502589419

Angle ∠ A = α = 101.81992285048° = 101°49'9″ = 1.77770807792 rad
Angle ∠ B = β = 66° = 1.15219173063 rad
Angle ∠ C = γ = 12.18107714952° = 12°10'51″ = 0.2132594568 rad

Height: ha = 8.86218640495
Height: hb = 9.49548543388
Height: hc = 41.11095455939

Median: ma = 20.5622101634
Median: mb = 24.87106659261
Median: mc = 43.25547683868

Inradius: r = 4.12439063756
Circumradius: R = 22.98773618486

Vertex coordinates: A[9.70105178838; 0] B[0; 0] C[18.30331489384; 41.11095455939]
Centroid: CG[9.33545556074; 13.70331818646]
Coordinates of the circumscribed circle: U[4.85502589419; 22.47698418543]
Coordinates of the inscribed circle: I[6.35502589419; 4.12439063756]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 78.18107714952° = 78°10'51″ = 1.77770807792 rad
∠ B' = β' = 114° = 1.15219173063 rad
∠ C' = γ' = 167.81992285048° = 167°49'9″ = 0.2132594568 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 45 ; ; b = 42 ; ; beta = 66° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 45**2 + c**2 -2 * 45 * c * cos (66° ) ; ; ; ; c**2 -36.606c +261 =0 ; ; p=1; q=-36.606; r=261 ; ; D = q**2 - 4pr = 36.606**2 - 4 * 1 * 261 = 296.02104424663 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 36.61 ± sqrt{ 296.02 } }{ 2 } ; ; c_{1,2} = 18.30314894 ± 8.6026310546051 ; ; c_{1} = 26.905779994605 ; ; : Nr. 1
c_{2} = 9.7005178853949 ; ; ; ; text{ Factored form: } ; ; (c -26.905779994605) (c -9.7005178853949) = 0 ; ; ; ; c>0 ; ; : Nr. 1


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 = 45 ; ; b = 42 ; ; c = 9.7 ; ;

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

p = a+b+c = 45+42+9.7 = 96.7 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 96.7 }{ 2 } = 48.35 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 48.35 * (48.35-45)(48.35-42)(48.35-9.7) } ; ; T = sqrt{ 39757.15 } = 199.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 199.39 }{ 45 } = 8.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 199.39 }{ 42 } = 9.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 199.39 }{ 9.7 } = 41.11 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 42**2+9.7**2-45**2 }{ 2 * 42 * 9.7 } ) = 101° 49'9" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+9.7**2-42**2 }{ 2 * 45 * 9.7 } ) = 66° ; ; gamma = 180° - alpha - beta = 180° - 101° 49'9" - 66° = 12° 10'51" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 199.39 }{ 48.35 } = 4.12 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45 }{ 2 * sin 101° 49'9" } = 22.99 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 9.7**2 - 45**2 } }{ 2 } = 20.562 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.7**2+2 * 45**2 - 42**2 } }{ 2 } = 24.871 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 45**2 - 9.7**2 } }{ 2 } = 43.255 ; ;
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