Triangle calculator SSA

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Triangle has two solutions with side c=79.04404993406 and with side c=10.1854652257

#1 Acute scalene triangle.

Sides: a = 61   b = 54   c = 79.04404993406

Area: T = 1644.117747333
Perimeter: p = 194.0440499341
Semiperimeter: s = 97.02202496703

Angle ∠ A = α = 50.3990321306° = 50°23'25″ = 0.87994770179 rad
Angle ∠ B = β = 43° = 0.75504915784 rad
Angle ∠ C = γ = 86.6109678694° = 86°36'35″ = 1.51216240573 rad

Height: ha = 53.90554909288
Height: hb = 60.89332397529
Height: hc = 41.60218999638

Median: ma = 60.42772311794
Median: mb = 65.23218960939
Median: mc = 41.91224070652

Inradius: r = 16.94661270087
Circumradius: R = 39.59895380123

Vertex coordinates: A[79.04404993406; 0] B[0; 0] C[44.61325757988; 41.60218999638]
Centroid: CG[41.21876917131; 13.86772999879]
Coordinates of the circumscribed circle: U[39.52202496703; 2.34112360032]
Coordinates of the inscribed circle: I[43.02202496703; 16.94661270087]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.6109678694° = 129°36'35″ = 0.87994770179 rad
∠ B' = β' = 137° = 0.75504915784 rad
∠ C' = γ' = 93.3990321306° = 93°23'25″ = 1.51216240573 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 61 ; ; b = 54 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 54**2 = 61**2 + c**2 -2 * 61 * c * cos (43° ) ; ; ; ; c**2 -89.225c +805 =0 ; ; p=1; q=-89.225; r=805 ; ; D = q**2 - 4pr = 89.225**2 - 4 * 1 * 805 = 4741.1276776 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 89.23 ± sqrt{ 4741.13 } }{ 2 } ; ; c_{1,2} = 44.6125758 ± 34.4279235418 ; ; c_{1} = 79.0404993418 ; ;
c_{2} = 10.1846522582 ; ; ; ; (c -79.0404993418) (c -10.1846522582) = 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 = 61 ; ; b = 54 ; ; c = 79.04 ; ;

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

p = a+b+c = 61+54+79.04 = 194.04 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 194.04 }{ 2 } = 97.02 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 97.02 * (97.02-61)(97.02-54)(97.02-79.04) } ; ; T = sqrt{ 2703122.27 } = 1644.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1644.12 }{ 61 } = 53.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1644.12 }{ 54 } = 60.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1644.12 }{ 79.04 } = 41.6 ; ;

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{ 61**2-54**2-79.04**2 }{ 2 * 54 * 79.04 } ) = 50° 23'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 54**2-61**2-79.04**2 }{ 2 * 61 * 79.04 } ) = 43° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 79.04**2-61**2-54**2 }{ 2 * 54 * 61 } ) = 86° 36'35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1644.12 }{ 97.02 } = 16.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 61 }{ 2 * sin 50° 23'25" } = 39.59 ; ;





#2 Obtuse scalene triangle.

Sides: a = 61   b = 54   c = 10.1854652257

Area: T = 211.855044218
Perimeter: p = 125.1854652257
Semiperimeter: s = 62.59223261285

Angle ∠ A = α = 129.6109678694° = 129°36'35″ = 2.26221156357 rad
Angle ∠ B = β = 43° = 0.75504915784 rad
Angle ∠ C = γ = 7.3990321306° = 7°23'25″ = 0.12989854396 rad

Height: ha = 6.94659161371
Height: hb = 7.84663126733
Height: hc = 41.60218999638

Median: ma = 24.07551650212
Median: mb = 34.44000519011
Median: mc = 57.38109046164

Inradius: r = 3.38546072719
Circumradius: R = 39.59895380123

Vertex coordinates: A[10.1854652257; 0] B[0; 0] C[44.61325757988; 41.60218999638]
Centroid: CG[18.26657426852; 13.86772999879]
Coordinates of the circumscribed circle: U[5.09223261285; 39.26106639606]
Coordinates of the inscribed circle: I[8.59223261285; 3.38546072719]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 50.3990321306° = 50°23'25″ = 2.26221156357 rad
∠ B' = β' = 137° = 0.75504915784 rad
∠ C' = γ' = 172.6109678694° = 172°36'35″ = 0.12989854396 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 61 ; ; b = 54 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 54**2 = 61**2 + c**2 -2 * 61 * c * cos (43° ) ; ; ; ; c**2 -89.225c +805 =0 ; ; p=1; q=-89.225; r=805 ; ; D = q**2 - 4pr = 89.225**2 - 4 * 1 * 805 = 4741.1276776 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 89.23 ± sqrt{ 4741.13 } }{ 2 } ; ; c_{1,2} = 44.6125758 ± 34.4279235418 ; ; c_{1} = 79.0404993418 ; ; : Nr. 1
c_{2} = 10.1846522582 ; ; ; ; (c -79.0404993418) (c -10.1846522582) = 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 = 61 ; ; b = 54 ; ; c = 10.18 ; ;

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

p = a+b+c = 61+54+10.18 = 125.18 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 125.18 }{ 2 } = 62.59 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.59 * (62.59-61)(62.59-54)(62.59-10.18) } ; ; T = sqrt{ 44880.61 } = 211.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 211.85 }{ 61 } = 6.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 211.85 }{ 54 } = 7.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 211.85 }{ 10.18 } = 41.6 ; ;

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{ 61**2-54**2-10.18**2 }{ 2 * 54 * 10.18 } ) = 129° 36'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 54**2-61**2-10.18**2 }{ 2 * 61 * 10.18 } ) = 43° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.18**2-61**2-54**2 }{ 2 * 54 * 61 } ) = 7° 23'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 211.85 }{ 62.59 } = 3.38 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 61 }{ 2 * sin 129° 36'35" } = 39.59 ; ;




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