Triangle calculator SSA

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Triangle has two solutions with side c=4.38109934777 and with side c=0.74986886289

#1 Acute scalene triangle.

Sides: a = 8.3   b = 8.1   c = 4.38109934777

Area: T = 17.29112754386
Perimeter: p = 20.78109934777
Semiperimeter: s = 10.39904967389

Angle ∠ A = α = 77.04331990626° = 77°2'36″ = 1.34546574899 rad
Angle ∠ B = β = 72° = 1.25766370614 rad
Angle ∠ C = γ = 30.95768009374° = 30°57'24″ = 0.54402981022 rad

Height: ha = 4.16765723949
Height: hb = 4.26994507256
Height: hc = 7.89437690852

Median: ma = 5.01878732473
Median: mb = 5.25772856044
Median: mc = 7.90326403206

Inradius: r = 1.6644143291
Circumradius: R = 4.25884220082

Vertex coordinates: A[4.38109934777; 0] B[0; 0] C[2.56548410533; 7.89437690852]
Centroid: CG[2.3155278177; 2.63112563617]
Coordinates of the circumscribed circle: U[2.19904967389; 3.65218326956]
Coordinates of the inscribed circle: I[2.29904967389; 1.6644143291]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102.9576800937° = 102°57'24″ = 1.34546574899 rad
∠ B' = β' = 108° = 1.25766370614 rad
∠ C' = γ' = 149.0433199063° = 149°2'36″ = 0.54402981022 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 = 8.3 ; ; b = 8.1 ; ; c = 4.38 ; ;

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

p = a+b+c = 8.3+8.1+4.38 = 20.78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.78 }{ 2 } = 10.39 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.39 * (10.39-8.3)(10.39-8.1)(10.39-4.38) } ; ; T = sqrt{ 298.99 } = 17.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.29 }{ 8.3 } = 4.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.29 }{ 8.1 } = 4.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.29 }{ 4.38 } = 7.89 ; ;

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{ 8.3**2-8.1**2-4.38**2 }{ 2 * 8.1 * 4.38 } ) = 77° 2'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.1**2-8.3**2-4.38**2 }{ 2 * 8.3 * 4.38 } ) = 72° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.38**2-8.3**2-8.1**2 }{ 2 * 8.1 * 8.3 } ) = 30° 57'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.29 }{ 10.39 } = 1.66 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.3 }{ 2 * sin 77° 2'36" } = 4.26 ; ;





#2 Obtuse scalene triangle.

Sides: a = 8.3   b = 8.1   c = 0.74986886289

Area: T = 2.95549875766
Perimeter: p = 17.14986886289
Semiperimeter: s = 8.57443443144

Angle ∠ A = α = 102.9576800937° = 102°57'24″ = 1.79769351637 rad
Angle ∠ B = β = 72° = 1.25766370614 rad
Angle ∠ C = γ = 5.04331990626° = 5°2'36″ = 0.08880204285 rad

Height: ha = 0.71220451992
Height: hb = 0.73296265621
Height: hc = 7.89437690852

Median: ma = 3.9832808975
Median: mb = 4.28105101719
Median: mc = 8.19220611774

Inradius: r = 0.34546313174
Circumradius: R = 4.25884220082

Vertex coordinates: A[0.74986886289; 0] B[0; 0] C[2.56548410533; 7.89437690852]
Centroid: CG[1.10545098941; 2.63112563617]
Coordinates of the circumscribed circle: U[0.37443443144; 4.24219363897]
Coordinates of the inscribed circle: I[0.47443443144; 0.34546313174]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 77.04331990626° = 77°2'36″ = 1.79769351637 rad
∠ B' = β' = 108° = 1.25766370614 rad
∠ C' = γ' = 174.9576800937° = 174°57'24″ = 0.08880204285 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 8.3 ; ; b = 8.1 ; ; beta = 72° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 8.1**2 = 8.3**2 + c**2 -2 * 8.1 * c * cos (72° ) ; ; ; ; c**2 -5.13c +3.28 =0 ; ; p=1; q=-5.12968210662; r=3.28 ; ; D = q**2 - 4pr = 5.13**2 - 4 * 1 * 3.28 = 13.193638515 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 5.13 ± sqrt{ 13.19 } }{ 2 } ; ; c_{1,2} = 2.56484105331 ± 1.81615242443 ; ; c_{1} = 4.38099347774 ; ;
c_{2} = 0.748688628884 ; ; ; ; (c -4.38099347774) (c -0.748688628884) = 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 = 8.3 ; ; b = 8.1 ; ; c = 0.75 ; ;

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

p = a+b+c = 8.3+8.1+0.75 = 17.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.15 }{ 2 } = 8.57 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.57 * (8.57-8.3)(8.57-8.1)(8.57-0.75) } ; ; T = sqrt{ 8.73 } = 2.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.95 }{ 8.3 } = 0.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.95 }{ 8.1 } = 0.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.95 }{ 0.75 } = 7.89 ; ;

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{ 8.3**2-8.1**2-0.75**2 }{ 2 * 8.1 * 0.75 } ) = 102° 57'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.1**2-8.3**2-0.75**2 }{ 2 * 8.3 * 0.75 } ) = 72° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.75**2-8.3**2-8.1**2 }{ 2 * 8.1 * 8.3 } ) = 5° 2'36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.95 }{ 8.57 } = 0.34 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.3 }{ 2 * sin 102° 57'24" } = 4.26 ; ;




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