Triangle calculator SSA

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Triangle has two solutions with side c=62.59329618281 and with side c=29.33223713462

#1 Acute scalene triangle.

Sides: a = 60   b = 42   c = 62.59329618281

Area: T = 1207.01994095
Perimeter: p = 164.5932961828
Semiperimeter: s = 82.2966480914

Angle ∠ A = α = 66.67441765214° = 66°40'27″ = 1.16436839064 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 73.32658234786° = 73°19'33″ = 1.28797770464 rad

Height: ha = 40.23439803167
Height: hb = 57.47771147381
Height: hc = 38.56772565812

Median: ma = 44.05660941892
Median: mb = 57.60215575762
Median: mc = 41.26217290282

Inradius: r = 14.66767195984
Circumradius: R = 32.67702003641

Vertex coordinates: A[62.59329618281; 0] B[0; 0] C[45.96326665871; 38.56772565812]
Centroid: CG[36.18552094717; 12.85657521937]
Coordinates of the circumscribed circle: U[31.2966480914; 9.3744021241]
Coordinates of the inscribed circle: I[40.2966480914; 14.66767195984]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 113.3265823479° = 113°19'33″ = 1.16436839064 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 106.6744176521° = 106°40'27″ = 1.28797770464 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 42 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 60**2 + c**2 -2 * 60 * c * cos (40° ) ; ; ; ; c**2 -91.925c +1836 =0 ; ; p=1; q=-91.925; r=1836 ; ; D = q**2 - 4pr = 91.925**2 - 4 * 1 * 1836 = 1106.2668792 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 91.93 ± sqrt{ 1106.27 } }{ 2 } ; ;
c_{1,2} = 45.96266659 ± 16.6302952409 ; ; c_{1} = 62.5929618281 ; ; c_{2} = 29.3323713462 ; ; ; ; text{ Factored form: } ; ; (c -62.5929618281) (c -29.3323713462) = 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 = 60 ; ; b = 42 ; ; c = 62.59 ; ;

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

p = a+b+c = 60+42+62.59 = 164.59 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 164.59 }{ 2 } = 82.3 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 82.3 * (82.3-60)(82.3-42)(82.3-62.59) } ; ; T = sqrt{ 1456895.85 } = 1207.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1207.02 }{ 60 } = 40.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1207.02 }{ 42 } = 57.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1207.02 }{ 62.59 } = 38.57 ; ;

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{ 42**2+62.59**2-60**2 }{ 2 * 42 * 62.59 } ) = 66° 40'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+62.59**2-42**2 }{ 2 * 60 * 62.59 } ) = 40° ; ;
 gamma = 180° - alpha - beta = 180° - 66° 40'27" - 40° = 73° 19'33" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1207.02 }{ 82.3 } = 14.67 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60 }{ 2 * sin 66° 40'27" } = 32.67 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 62.59**2 - 60**2 } }{ 2 } = 44.056 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 62.59**2+2 * 60**2 - 42**2 } }{ 2 } = 57.602 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 60**2 - 62.59**2 } }{ 2 } = 41.262 ; ;



#2 Obtuse scalene triangle.

Sides: a = 60   b = 42   c = 29.33223713462

Area: T = 565.6354545922
Perimeter: p = 131.3322371346
Semiperimeter: s = 65.66661856731

Angle ∠ A = α = 113.3265823479° = 113°19'33″ = 1.97879087472 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 26.67441765214° = 26°40'27″ = 0.46655522056 rad

Height: ha = 18.85444848641
Height: hb = 26.93549783772
Height: hc = 38.56772565812

Median: ma = 20.3032561523
Median: mb = 42.2998865285
Median: mc = 49.66879272549

Inradius: r = 8.61437871436
Circumradius: R = 32.67702003641

Vertex coordinates: A[29.33223713462; 0] B[0; 0] C[45.96326665871; 38.56772565812]
Centroid: CG[25.09883459778; 12.85657521937]
Coordinates of the circumscribed circle: U[14.66661856731; 29.19332353402]
Coordinates of the inscribed circle: I[23.66661856731; 8.61437871436]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 66.67441765214° = 66°40'27″ = 1.97879087472 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 153.3265823479° = 153°19'33″ = 0.46655522056 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 42 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 60**2 + c**2 -2 * 60 * c * cos (40° ) ; ; ; ; c**2 -91.925c +1836 =0 ; ; p=1; q=-91.925; r=1836 ; ; D = q**2 - 4pr = 91.925**2 - 4 * 1 * 1836 = 1106.2668792 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 91.93 ± sqrt{ 1106.27 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 45.96266659 ± 16.6302952409 ; ; c_{1} = 62.5929618281 ; ; c_{2} = 29.3323713462 ; ; ; ; text{ Factored form: } ; ; (c -62.5929618281) (c -29.3323713462) = 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 = 60 ; ; b = 42 ; ; c = 29.33 ; ;

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

p = a+b+c = 60+42+29.33 = 131.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 131.33 }{ 2 } = 65.67 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.67 * (65.67-60)(65.67-42)(65.67-29.33) } ; ; T = sqrt{ 319942.44 } = 565.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 565.63 }{ 60 } = 18.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 565.63 }{ 42 } = 26.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 565.63 }{ 29.33 } = 38.57 ; ;

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{ 42**2+29.33**2-60**2 }{ 2 * 42 * 29.33 } ) = 113° 19'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+29.33**2-42**2 }{ 2 * 60 * 29.33 } ) = 40° ; ;
 gamma = 180° - alpha - beta = 180° - 113° 19'33" - 40° = 26° 40'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 565.63 }{ 65.67 } = 8.61 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60 }{ 2 * sin 113° 19'33" } = 32.67 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 29.33**2 - 60**2 } }{ 2 } = 20.303 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.33**2+2 * 60**2 - 42**2 } }{ 2 } = 42.299 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 60**2 - 29.33**2 } }{ 2 } = 49.668 ; ;
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