Triangle calculator SSA

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Triangle has two solutions with side c=126.3888321747 and with side c=15.03330344904

#1 Acute scalene triangle.

Sides: a = 100   b = 90   c = 126.3888321747

Area: T = 4468.50219685
Perimeter: p = 316.3888321747
Semiperimeter: s = 158.1944160874

Angle ∠ A = α = 51.78330767038° = 51°46'59″ = 0.90437851853 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 83.21769232962° = 83°13'1″ = 1.45224093049 rad

Height: ha = 89.37700393701
Height: hb = 99.33000437445
Height: hc = 70.71106781187

Median: ma = 97.65875851483
Median: mb = 104.7699589001
Median: mc = 71.10990573099

Inradius: r = 28.24769463084
Circumradius: R = 63.64396103068

Vertex coordinates: A[126.3888321747; 0] B[0; 0] C[70.71106781187; 70.71106781187]
Centroid: CG[65.76996666219; 23.57702260396]
Coordinates of the circumscribed circle: U[63.19441608735; 7.51765172452]
Coordinates of the inscribed circle: I[68.19441608735; 28.24769463084]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.2176923296° = 128°13'1″ = 0.90437851853 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 96.78330767038° = 96°46'59″ = 1.45224093049 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (45° ) ; ; ; ; c**2 -141.421c +1900 =0 ; ; p=1; q=-141.421; r=1900 ; ; D = q**2 - 4pr = 141.421**2 - 4 * 1 * 1900 = 12400 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 141.42 ± sqrt{ 12400 } }{ 2 } = fraction{ 141.42 ± 20 sqrt{ 31 } }{ 2 } ; ; c_{1,2} = 70.71067812 ± 55.6776436283 ; ; c_{1} = 126.388321748 ; ; c_{2} = 15.0330344917 ; ; ; ; text{ Factored form: } ; ; (c -126.388321748) (c -15.0330344917) = 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 = 100 ; ; b = 90 ; ; c = 126.39 ; ;

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

p = a+b+c = 100+90+126.39 = 316.39 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 316.39 }{ 2 } = 158.19 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 158.19 * (158.19-100)(158.19-90)(158.19-126.39) } ; ; T = sqrt{ 19967509.84 } = 4468.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4468.5 }{ 100 } = 89.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4468.5 }{ 90 } = 99.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4468.5 }{ 126.39 } = 70.71 ; ;

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{ 90**2+126.39**2-100**2 }{ 2 * 90 * 126.39 } ) = 51° 46'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+126.39**2-90**2 }{ 2 * 100 * 126.39 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 51° 46'59" - 45° = 83° 13'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4468.5 }{ 158.19 } = 28.25 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 51° 46'59" } = 63.64 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 126.39**2 - 100**2 } }{ 2 } = 97.658 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 126.39**2+2 * 100**2 - 90**2 } }{ 2 } = 104.7 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 126.39**2 } }{ 2 } = 71.109 ; ;







#2 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 15.03330344904

Area: T = 531.4988031497
Perimeter: p = 205.033303449
Semiperimeter: s = 102.5176517245

Angle ∠ A = α = 128.2176923296° = 128°13'1″ = 2.23878074683 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 6.78330767038° = 6°46'59″ = 0.11883870219 rad

Height: ha = 10.63299606299
Height: hb = 11.81110673666
Height: hc = 70.71106781187

Median: ma = 40.78798487368
Median: mb = 55.57697405338
Median: mc = 94.8344075988

Inradius: r = 5.18545111966
Circumradius: R = 63.64396103068

Vertex coordinates: A[15.03330344904; 0] B[0; 0] C[70.71106781187; 70.71106781187]
Centroid: CG[28.58112375363; 23.57702260396]
Coordinates of the circumscribed circle: U[7.51765172452; 63.19441608735]
Coordinates of the inscribed circle: I[12.51765172452; 5.18545111966]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 51.78330767038° = 51°46'59″ = 2.23878074683 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 173.2176923296° = 173°13'1″ = 0.11883870219 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (45° ) ; ; ; ; c**2 -141.421c +1900 =0 ; ; p=1; q=-141.421; r=1900 ; ; D = q**2 - 4pr = 141.421**2 - 4 * 1 * 1900 = 12400 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 141.42 ± sqrt{ 12400 } }{ 2 } = fraction{ 141.42 ± 20 sqrt{ 31 } }{ 2 } ; ; c_{1,2} = 70.71067812 ± 55.6776436283 ; ; c_{1} = 126.388321748 ; ; c_{2} = 15.0330344917 ; ; ; ; text{ Factored form: } ; ; (c -126.388321748) (c -15.0330344917) = 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 = 100 ; ; b = 90 ; ; c = 15.03 ; ;

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

p = a+b+c = 100+90+15.03 = 205.03 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 205.03 }{ 2 } = 102.52 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.52 * (102.52-100)(102.52-90)(102.52-15.03) } ; ; T = sqrt{ 282490.16 } = 531.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 531.5 }{ 100 } = 10.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 531.5 }{ 90 } = 11.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 531.5 }{ 15.03 } = 70.71 ; ;

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{ 90**2+15.03**2-100**2 }{ 2 * 90 * 15.03 } ) = 128° 13'1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+15.03**2-90**2 }{ 2 * 100 * 15.03 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 128° 13'1" - 45° = 6° 46'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 531.5 }{ 102.52 } = 5.18 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 128° 13'1" } = 63.64 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 15.03**2 - 100**2 } }{ 2 } = 40.78 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.03**2+2 * 100**2 - 90**2 } }{ 2 } = 55.57 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 15.03**2 } }{ 2 } = 94.834 ; ;
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