Triangle calculator SSA

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Triangle has two solutions with side c=1.93218516526 and with side c=0.51876380902

#1 Acute scalene triangle.

Sides: a = 1.73220508076   b = 1.41442135624   c = 1.93218516526

Area: T = 1.18330127019
Perimeter: p = 5.07881160225
Semiperimeter: s = 2.53990580113

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 75° = 1.3098996939 rad

Height: ha = 1.36660254038
Height: hb = 1.67330326075
Height: hc = 1.22547448714

Median: ma = 1.45546564556
Median: mb = 1.69329339632
Median: mc = 1.25217936324

Inradius: r = 0.46659258263
Circumradius: R = 1

Vertex coordinates: A[1.93218516526; 0] B[0; 0] C[1.22547448714; 1.22547448714]
Centroid: CG[1.05221988413; 0.40882482905]
Coordinates of the circumscribed circle: U[0.96659258263; 0.25988190451]
Coordinates of the inscribed circle: I[1.12548444489; 0.46659258263]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 105° = 1.3098996939 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 1.73 ; ; b = 1.41 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 1.41**2 = 1.73**2 + c**2 -2 * 1.73 * c * cos (45° ) ; ; ; ; c**2 -2.449c +1 =0 ; ; p=1; q=-2.449; r=1 ; ; D = q**2 - 4pr = 2.449**2 - 4 * 1 * 1 = 2 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 2.45 ± sqrt{ 2 } }{ 2 } ; ; c_{1,2} = 1.22474487 ± 0.707106781187 ; ; c_{1} = 1.93185165119 ; ; c_{2} = 0.517638088813 ; ; ; ; text{ Factored form: } ; ; (c -1.93185165119) (c -0.517638088813) = 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 = 1.73 ; ; b = 1.41 ; ; c = 1.93 ; ;

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

p = a+b+c = 1.73+1.41+1.93 = 5.08 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5.08 }{ 2 } = 2.54 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.54 * (2.54-1.73)(2.54-1.41)(2.54-1.93) } ; ; T = sqrt{ 1.4 } = 1.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.18 }{ 1.73 } = 1.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.18 }{ 1.41 } = 1.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.18 }{ 1.93 } = 1.22 ; ;

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{ 1.41**2+1.93**2-1.73**2 }{ 2 * 1.41 * 1.93 } ) = 60° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.73**2+1.93**2-1.41**2 }{ 2 * 1.73 * 1.93 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 60° - 45° = 75° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.18 }{ 2.54 } = 0.47 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.73 }{ 2 * sin 60° } = 1 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 1.93**2 - 1.73**2 } }{ 2 } = 1.455 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.93**2+2 * 1.73**2 - 1.41**2 } }{ 2 } = 1.693 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 1.73**2 - 1.93**2 } }{ 2 } = 1.252 ; ;







#2 Obtuse scalene triangle.

Sides: a = 1.73220508076   b = 1.41442135624   c = 0.51876380902

Area: T = 0.31769872981
Perimeter: p = 3.66439024601
Semiperimeter: s = 1.83219512301

Angle ∠ A = α = 120° = 2.09443951024 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 15° = 0.26217993878 rad

Height: ha = 0.36660254038
Height: hb = 0.44882877361
Height: hc = 1.22547448714

Median: ma = 0.62196568375
Median: mb = 1.0654882433
Median: mc = 1.56598117521

Inradius: r = 0.17330326075
Circumradius: R = 1

Vertex coordinates: A[0.51876380902; 0] B[0; 0] C[1.22547448714; 1.22547448714]
Centroid: CG[0.58107943205; 0.40882482905]
Coordinates of the circumscribed circle: U[0.25988190451; 0.96659258263]
Coordinates of the inscribed circle: I[0.41877376677; 0.17330326075]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 60° = 2.09443951024 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 165° = 0.26217993878 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 1.73 ; ; b = 1.41 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 1.41**2 = 1.73**2 + c**2 -2 * 1.73 * c * cos (45° ) ; ; ; ; c**2 -2.449c +1 =0 ; ; p=1; q=-2.449; r=1 ; ; D = q**2 - 4pr = 2.449**2 - 4 * 1 * 1 = 2 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 2.45 ± sqrt{ 2 } }{ 2 } ; ; c_{1,2} = 1.22474487 ± 0.707106781187 ; ; c_{1} = 1.93185165119 ; ; c_{2} = 0.517638088813 ; ; ; ; text{ Factored form: } ; ; (c -1.93185165119) (c -0.517638088813) = 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 = 1.73 ; ; b = 1.41 ; ; c = 0.52 ; ;

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

p = a+b+c = 1.73+1.41+0.52 = 3.66 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 3.66 }{ 2 } = 1.83 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.83 * (1.83-1.73)(1.83-1.41)(1.83-0.52) } ; ; T = sqrt{ 0.1 } = 0.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.32 }{ 1.73 } = 0.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.32 }{ 1.41 } = 0.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.32 }{ 0.52 } = 1.22 ; ;

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{ 1.41**2+0.52**2-1.73**2 }{ 2 * 1.41 * 0.52 } ) = 120° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.73**2+0.52**2-1.41**2 }{ 2 * 1.73 * 0.52 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 120° - 45° = 15° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.32 }{ 1.83 } = 0.17 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.73 }{ 2 * sin 120° } = 1 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 0.52**2 - 1.73**2 } }{ 2 } = 0.62 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.52**2+2 * 1.73**2 - 1.41**2 } }{ 2 } = 1.065 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 1.73**2 - 0.52**2 } }{ 2 } = 1.56 ; ;
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