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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 42**2 = 60**2 + c**2 -2 * 42 * c * cos (40° ) ; ; ; ; c**2 -91.925c +1836 =0 ; ; p=1; q=-91.9253331743; 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} = 45.9626665871 ± 16.6302952409 ; ; c_{1} = 62.5929618281 ; ;
c_{2} = 29.3323713462 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 60**2-42**2-29.33**2 }{ 2 * 42 * 29.33 } ) = 113° 19'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-60**2-29.33**2 }{ 2 * 60 * 29.33 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29.33**2-60**2-42**2 }{ 2 * 42 * 60 } ) = 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 ; ;




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