Triangle calculator SSA

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Triangle has two solutions with side c=79.78436813757 and with side c=29.81881277045

#1 Acute scalene triangle.

Sides: a = 98   b = 85   c = 79.78436813757

Area: T = 3241.043980732
Perimeter: p = 262.7843681376
Semiperimeter: s = 131.3921840688

Angle ∠ A = α = 72.90875112757° = 72°54'27″ = 1.27224761212 rad
Angle ∠ B = β = 56° = 0.97773843811 rad
Angle ∠ C = γ = 51.09224887243° = 51°5'33″ = 0.89217321513 rad

Height: ha = 66.14436695372
Height: hb = 76.26597601723
Height: hc = 81.24656821104

Median: ma = 66.28988973127
Median: mb = 78.60332308937
Median: mc = 82.60223065449

Inradius: r = 24.66769792459
Circumradius: R = 51.26442628114

Vertex coordinates: A[79.78436813757; 0] B[0; 0] C[54.80109045401; 81.24656821104]
Centroid: CG[44.86215286386; 27.08218940368]
Coordinates of the circumscribed circle: U[39.89218406879; 32.19772931802]
Coordinates of the inscribed circle: I[46.39218406879; 24.66769792459]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.0922488724° = 107°5'33″ = 1.27224761212 rad
∠ B' = β' = 124° = 0.97773843811 rad
∠ C' = γ' = 128.9087511276° = 128°54'27″ = 0.89217321513 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 98 ; ; b = 85 ; ; beta = 56° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 85**2 = 98**2 + c**2 -2 * 98 * c * cos (56° ) ; ; ; ; c**2 -109.602c +2379 =0 ; ; p=1; q=-109.602; r=2379 ; ; D = q**2 - 4pr = 109.602**2 - 4 * 1 * 2379 = 2496.55655367 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 109.6 ± sqrt{ 2496.56 } }{ 2 } ; ; c_{1,2} = 54.80090454 ± 24.9827768356 ; ; c_{1} = 79.7836813756 ; ; c_{2} = 29.8181277044 ; ; ; ; text{ Factored form: } ; ; (c -79.7836813756) (c -29.8181277044) = 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 = 98 ; ; b = 85 ; ; c = 79.78 ; ;

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

p = a+b+c = 98+85+79.78 = 262.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 262.78 }{ 2 } = 131.39 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 131.39 * (131.39-98)(131.39-85)(131.39-79.78) } ; ; T = sqrt{ 10504339.03 } = 3241.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3241.04 }{ 98 } = 66.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3241.04 }{ 85 } = 76.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3241.04 }{ 79.78 } = 81.25 ; ;

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{ 85**2+79.78**2-98**2 }{ 2 * 85 * 79.78 } ) = 72° 54'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 98**2+79.78**2-85**2 }{ 2 * 98 * 79.78 } ) = 56° ; ; gamma = 180° - alpha - beta = 180° - 72° 54'27" - 56° = 51° 5'33" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3241.04 }{ 131.39 } = 24.67 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 98 }{ 2 * sin 72° 54'27" } = 51.26 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 85**2+2 * 79.78**2 - 98**2 } }{ 2 } = 66.289 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 79.78**2+2 * 98**2 - 85**2 } }{ 2 } = 78.603 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 85**2+2 * 98**2 - 79.78**2 } }{ 2 } = 82.602 ; ;







#2 Obtuse scalene triangle.

Sides: a = 98   b = 85   c = 29.81881277045

Area: T = 1211.297706231
Perimeter: p = 212.8188127705
Semiperimeter: s = 106.4099063852

Angle ∠ A = α = 107.0922488724° = 107°5'33″ = 1.86991165324 rad
Angle ∠ B = β = 56° = 0.97773843811 rad
Angle ∠ C = γ = 16.90875112757° = 16°54'27″ = 0.29550917401 rad

Height: ha = 24.72203482103
Height: hb = 28.50111073484
Height: hc = 81.24656821104

Median: ma = 40.69547216467
Median: mb = 58.65441590162
Median: mc = 90.51108823018

Inradius: r = 11.38334011733
Circumradius: R = 51.26442628114

Vertex coordinates: A[29.81881277045; 0] B[0; 0] C[54.80109045401; 81.24656821104]
Centroid: CG[28.20663440816; 27.08218940368]
Coordinates of the circumscribed circle: U[14.90990638523; 49.04883889302]
Coordinates of the inscribed circle: I[21.40990638523; 11.38334011733]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.90875112757° = 72°54'27″ = 1.86991165324 rad
∠ B' = β' = 124° = 0.97773843811 rad
∠ C' = γ' = 163.0922488724° = 163°5'33″ = 0.29550917401 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 98 ; ; b = 85 ; ; beta = 56° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 85**2 = 98**2 + c**2 -2 * 98 * c * cos (56° ) ; ; ; ; c**2 -109.602c +2379 =0 ; ; p=1; q=-109.602; r=2379 ; ; D = q**2 - 4pr = 109.602**2 - 4 * 1 * 2379 = 2496.55655367 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 109.6 ± sqrt{ 2496.56 } }{ 2 } ; ; c_{1,2} = 54.80090454 ± 24.9827768356 ; ; c_{1} = 79.7836813756 ; ; c_{2} = 29.8181277044 ; ; ; ; text{ Factored form: } ; ; (c -79.7836813756) (c -29.8181277044) = 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 = 98 ; ; b = 85 ; ; c = 29.82 ; ;

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

p = a+b+c = 98+85+29.82 = 212.82 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 212.82 }{ 2 } = 106.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 106.41 * (106.41-98)(106.41-85)(106.41-29.82) } ; ; T = sqrt{ 1467240.57 } = 1211.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1211.3 }{ 98 } = 24.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1211.3 }{ 85 } = 28.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1211.3 }{ 29.82 } = 81.25 ; ;

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{ 85**2+29.82**2-98**2 }{ 2 * 85 * 29.82 } ) = 107° 5'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 98**2+29.82**2-85**2 }{ 2 * 98 * 29.82 } ) = 56° ; ; gamma = 180° - alpha - beta = 180° - 107° 5'33" - 56° = 16° 54'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1211.3 }{ 106.41 } = 11.38 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 98 }{ 2 * sin 107° 5'33" } = 51.26 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 85**2+2 * 29.82**2 - 98**2 } }{ 2 } = 40.695 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.82**2+2 * 98**2 - 85**2 } }{ 2 } = 58.654 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 85**2+2 * 98**2 - 29.82**2 } }{ 2 } = 90.511 ; ;
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