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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ 1.73**2-1.41**2-1.93**2 }{ 2 * 1.41 * 1.93 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.41**2-1.73**2-1.93**2 }{ 2 * 1.73 * 1.93 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.93**2-1.73**2-1.41**2 }{ 2 * 1.41 * 1.73 } ) = 75° ; ;

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 1.41**2 = 1.73**2 + c**2 -2 * 1.41 * c * cos (45° ) ; ; ; ; c**2 -2.449c +1 =0 ; ; p=1; q=-2.44948974278; 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} = 1.22474487139 ± 0.707106781187 ; ; c_{1} = 1.93185165258 ; ;
c_{2} = 0.517638090205 ; ; ; ; (c -1.93185165258) (c -0.517638090205) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 1.73**2-1.41**2-0.52**2 }{ 2 * 1.41 * 0.52 } ) = 120° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.41**2-1.73**2-0.52**2 }{ 2 * 1.73 * 0.52 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.52**2-1.73**2-1.41**2 }{ 2 * 1.41 * 1.73 } ) = 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 ; ;




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