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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 54**2 = 61**2 + c**2 -2 * 54 * c * cos (43° ) ; ; ; ; c**2 -89.225c +805 =0 ; ; p=1; q=-89.2251515975; 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} = 44.6125757988 ± 34.4279235418 ; ; c_{1} = 79.0404993406 ; ;
c_{2} = 10.184652257 ; ; ; ; (c -79.0404993406) (c -10.184652257) = 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 = 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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