Triangle calculator SSA

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Triangle has two solutions with side c=26.29216650122 and with side c=7.6455008405

#1 Acute scalene triangle.

Sides: a = 35   b = 32   c = 26.29216650122

Area: T = 402.4166146181
Perimeter: p = 93.29216650122
Semiperimeter: s = 46.64658325061

Angle ∠ A = α = 73.06109640073° = 73°3'39″ = 1.27551543766 rad
Angle ∠ B = β = 61° = 1.06546508437 rad
Angle ∠ C = γ = 45.93990359927° = 45°56'21″ = 0.80217874333 rad

Height: ha = 22.99552083532
Height: hb = 25.15110091363
Height: hc = 30.61216897499

Median: ma = 23.48113931562
Median: mb = 26.49876569636
Median: mc = 30.84994260517

Inradius: r = 8.62770546491
Circumradius: R = 18.2943665086

Vertex coordinates: A[26.29216650122; 0] B[0; 0] C[16.96883367086; 30.61216897499]
Centroid: CG[14.42200005736; 10.20438965833]
Coordinates of the circumscribed circle: U[13.14658325061; 12.72218422408]
Coordinates of the inscribed circle: I[14.64658325061; 8.62770546491]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 106.9399035993° = 106°56'21″ = 1.27551543766 rad
∠ B' = β' = 119° = 1.06546508437 rad
∠ C' = γ' = 134.0610964007° = 134°3'39″ = 0.80217874333 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 32 ; ; beta = 61° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32**2 = 35**2 + c**2 -2 * 35 * c * cos (61° ) ; ; ; ; c**2 -33.937c +201 =0 ; ; p=1; q=-33.937; r=201 ; ; D = q**2 - 4pr = 33.937**2 - 4 * 1 * 201 = 347.697802629 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 33.94 ± sqrt{ 347.7 } }{ 2 } ; ; c_{1,2} = 16.96833671 ± 9.32332830362 ; ; c_{1} = 26.2916650136 ; ;
c_{2} = 7.64500840638 ; ; ; ; text{ Factored form: } ; ; (c -26.2916650136) (c -7.64500840638) = 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 = 35 ; ; b = 32 ; ; c = 26.29 ; ;

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

p = a+b+c = 35+32+26.29 = 93.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 93.29 }{ 2 } = 46.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 46.65 * (46.65-35)(46.65-32)(46.65-26.29) } ; ; T = sqrt{ 161938.75 } = 402.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 402.42 }{ 35 } = 23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 402.42 }{ 32 } = 25.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 402.42 }{ 26.29 } = 30.61 ; ;

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{ 32**2+26.29**2-35**2 }{ 2 * 32 * 26.29 } ) = 73° 3'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+26.29**2-32**2 }{ 2 * 35 * 26.29 } ) = 61° ; ; gamma = 180° - alpha - beta = 180° - 73° 3'39" - 61° = 45° 56'21" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 402.42 }{ 46.65 } = 8.63 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 73° 3'39" } = 18.29 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 26.29**2 - 35**2 } }{ 2 } = 23.481 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.29**2+2 * 35**2 - 32**2 } }{ 2 } = 26.498 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 35**2 - 26.29**2 } }{ 2 } = 30.849 ; ;







#2 Obtuse scalene triangle.

Sides: a = 35   b = 32   c = 7.6455008405

Area: T = 117.0133312715
Perimeter: p = 74.6455008405
Semiperimeter: s = 37.32325042025

Angle ∠ A = α = 106.9399035993° = 106°56'21″ = 1.8666438277 rad
Angle ∠ B = β = 61° = 1.06546508437 rad
Angle ∠ C = γ = 12.06109640073° = 12°3'39″ = 0.21105035329 rad

Height: ha = 6.68664750123
Height: hb = 7.31333320447
Height: hc = 30.61216897499

Median: ma = 15.32988315522
Median: mb = 19.64398339289
Median: mc = 33.3154988543

Inradius: r = 3.13551945754
Circumradius: R = 18.2943665086

Vertex coordinates: A[7.6455008405; 0] B[0; 0] C[16.96883367086; 30.61216897499]
Centroid: CG[8.20444483712; 10.20438965833]
Coordinates of the circumscribed circle: U[3.82325042025; 17.89898475091]
Coordinates of the inscribed circle: I[5.32325042025; 3.13551945754]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 73.06109640073° = 73°3'39″ = 1.8666438277 rad
∠ B' = β' = 119° = 1.06546508437 rad
∠ C' = γ' = 167.9399035993° = 167°56'21″ = 0.21105035329 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 32 ; ; beta = 61° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32**2 = 35**2 + c**2 -2 * 35 * c * cos (61° ) ; ; ; ; c**2 -33.937c +201 =0 ; ; p=1; q=-33.937; r=201 ; ; D = q**2 - 4pr = 33.937**2 - 4 * 1 * 201 = 347.697802629 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 33.94 ± sqrt{ 347.7 } }{ 2 } ; ; c_{1,2} = 16.96833671 ± 9.32332830362 ; ; c_{1} = 26.2916650136 ; ; : Nr. 1
c_{2} = 7.64500840638 ; ; ; ; text{ Factored form: } ; ; (c -26.2916650136) (c -7.64500840638) = 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 = 35 ; ; b = 32 ; ; c = 7.65 ; ;

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

p = a+b+c = 35+32+7.65 = 74.65 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74.65 }{ 2 } = 37.32 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.32 * (37.32-35)(37.32-32)(37.32-7.65) } ; ; T = sqrt{ 13692.12 } = 117.01 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 117.01 }{ 35 } = 6.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 117.01 }{ 32 } = 7.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 117.01 }{ 7.65 } = 30.61 ; ;

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{ 32**2+7.65**2-35**2 }{ 2 * 32 * 7.65 } ) = 106° 56'21" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+7.65**2-32**2 }{ 2 * 35 * 7.65 } ) = 61° ; ; gamma = 180° - alpha - beta = 180° - 106° 56'21" - 61° = 12° 3'39" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 117.01 }{ 37.32 } = 3.14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 106° 56'21" } = 18.29 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 7.65**2 - 35**2 } }{ 2 } = 15.329 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.65**2+2 * 35**2 - 32**2 } }{ 2 } = 19.64 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 35**2 - 7.65**2 } }{ 2 } = 33.315 ; ;
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