Triangle calculator SSA

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Triangle has two solutions with side c=79.78436813757 and with side c=29.81881277045

#1 Acute scalene triangle.

Sides: a = 98   b = 85   c = 79.78436813757

Area: T = 3241.043980732
Perimeter: p = 262.7843681376
Semiperimeter: s = 131.3921840688

Angle ∠ A = α = 72.90875112757° = 72°54'27″ = 1.27224761212 rad
Angle ∠ B = β = 56° = 0.97773843811 rad
Angle ∠ C = γ = 51.09224887243° = 51°5'33″ = 0.89217321513 rad

Height: ha = 66.14436695372
Height: hb = 76.26597601723
Height: hc = 81.24656821104

Median: ma = 66.28988973127
Median: mb = 78.60332308937
Median: mc = 82.60223065449

Inradius: r = 24.66769792459
Circumradius: R = 51.26442628114

Vertex coordinates: A[79.78436813757; 0] B[0; 0] C[54.80109045401; 81.24656821104]
Centroid: CG[44.86215286386; 27.08218940368]
Coordinates of the circumscribed circle: U[39.89218406879; 32.19772931802]
Coordinates of the inscribed circle: I[46.39218406879; 24.66769792459]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.0922488724° = 107°5'33″ = 1.27224761212 rad
∠ B' = β' = 124° = 0.97773843811 rad
∠ C' = γ' = 128.9087511276° = 128°54'27″ = 0.89217321513 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 = 98 ; ; b = 85 ; ; c = 79.78 ; ;

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

p = a+b+c = 98+85+79.78 = 262.78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 262.78 }{ 2 } = 131.39 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 131.39 * (131.39-98)(131.39-85)(131.39-79.78) } ; ; T = sqrt{ 10504339.03 } = 3241.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3241.04 }{ 98 } = 66.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3241.04 }{ 85 } = 76.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3241.04 }{ 79.78 } = 81.25 ; ;

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{ 98**2-85**2-79.78**2 }{ 2 * 85 * 79.78 } ) = 72° 54'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 85**2-98**2-79.78**2 }{ 2 * 98 * 79.78 } ) = 56° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 79.78**2-98**2-85**2 }{ 2 * 85 * 98 } ) = 51° 5'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3241.04 }{ 131.39 } = 24.67 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 98 }{ 2 * sin 72° 54'27" } = 51.26 ; ;





#2 Obtuse scalene triangle.

Sides: a = 98   b = 85   c = 29.81881277045

Area: T = 1211.297706231
Perimeter: p = 212.8188127705
Semiperimeter: s = 106.4099063852

Angle ∠ A = α = 107.0922488724° = 107°5'33″ = 1.86991165324 rad
Angle ∠ B = β = 56° = 0.97773843811 rad
Angle ∠ C = γ = 16.90875112757° = 16°54'27″ = 0.29550917401 rad

Height: ha = 24.72203482103
Height: hb = 28.50111073484
Height: hc = 81.24656821104

Median: ma = 40.69547216467
Median: mb = 58.65441590162
Median: mc = 90.51108823018

Inradius: r = 11.38334011733
Circumradius: R = 51.26442628114

Vertex coordinates: A[29.81881277045; 0] B[0; 0] C[54.80109045401; 81.24656821104]
Centroid: CG[28.20663440816; 27.08218940368]
Coordinates of the circumscribed circle: U[14.90990638523; 49.04883889302]
Coordinates of the inscribed circle: I[21.40990638523; 11.38334011733]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.90875112757° = 72°54'27″ = 1.86991165324 rad
∠ B' = β' = 124° = 0.97773843811 rad
∠ C' = γ' = 163.0922488724° = 163°5'33″ = 0.29550917401 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 98 ; ; b = 85 ; ; beta = 56° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 85**2 = 98**2 + c**2 -2 * 85 * c * cos (56° ) ; ; ; ; c**2 -109.602c +2379 =0 ; ; p=1; q=-109.60180908; r=2379 ; ; D = q**2 - 4pr = 109.602**2 - 4 * 1 * 2379 = 2496.55655367 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 109.6 ± sqrt{ 2496.56 } }{ 2 } ; ; c_{1,2} = 54.8009045401 ± 24.9827768356 ; ; c_{1} = 79.7836813757 ; ;
c_{2} = 29.8181277045 ; ; ; ; (c -79.7836813757) (c -29.8181277045) = 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 = 98 ; ; b = 85 ; ; c = 29.82 ; ;

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

p = a+b+c = 98+85+29.82 = 212.82 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 212.82 }{ 2 } = 106.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 106.41 * (106.41-98)(106.41-85)(106.41-29.82) } ; ; T = sqrt{ 1467240.57 } = 1211.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1211.3 }{ 98 } = 24.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1211.3 }{ 85 } = 28.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1211.3 }{ 29.82 } = 81.25 ; ;

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{ 98**2-85**2-29.82**2 }{ 2 * 85 * 29.82 } ) = 107° 5'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 85**2-98**2-29.82**2 }{ 2 * 98 * 29.82 } ) = 56° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29.82**2-98**2-85**2 }{ 2 * 85 * 98 } ) = 16° 54'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1211.3 }{ 106.41 } = 11.38 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 98 }{ 2 * sin 107° 5'33" } = 51.26 ; ;




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