Triangle calculator SSA

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Triangle has two solutions with side c=257.7990148494 and with side c=105.2476535442

#1 Acute scalene triangle.

Sides: a = 248.6   b = 186.2   c = 257.7990148494

Area: T = 21894.35770741
Perimeter: p = 692.5990148494
Semiperimeter: s = 346.2955074247

Angle ∠ A = α = 65.81988442228° = 65°49'8″ = 1.14987555415 rad
Angle ∠ B = β = 43.1° = 43°6' = 0.75222369076 rad
Angle ∠ C = γ = 71.08111557772° = 71°4'52″ = 1.24106002044 rad

Height: ha = 176.1411247579
Height: hb = 235.177032303
Height: hc = 169.8621860137

Median: ma = 187.3843591412
Median: mb = 235.5022123834
Median: mc = 177.8266488001

Inradius: r = 63.22545697451
Circumradius: R = 136.2565778556

Vertex coordinates: A[257.7990148494; 0] B[0; 0] C[181.5188341968; 169.8621860137]
Centroid: CG[146.4366163487; 56.62106200457]
Coordinates of the circumscribed circle: U[128.8955074247; 44.17880151753]
Coordinates of the inscribed circle: I[160.0955074247; 63.22545697451]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 114.1811155777° = 114°10'52″ = 1.14987555415 rad
∠ B' = β' = 136.9° = 136°54' = 0.75222369076 rad
∠ C' = γ' = 108.9198844223° = 108°55'8″ = 1.24106002044 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 248.6 ; ; b = 186.2 ; ; beta = 43° 6' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 186.2**2 = 248.6**2 + c**2 -2 * 248.6 * c * cos (43° 6') ; ; ; ; c**2 -363.037c +27131.52 =0 ; ; p=1; q=-363.037; r=27131.52 ; ; D = q**2 - 4pr = 363.037**2 - 4 * 1 * 27131.52 = 23269.5538831 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 363.04 ± sqrt{ 23269.55 } }{ 2 } ; ; c_{1,2} = 181.51834197 ± 76.2718065263 ; ;
c_{1} = 257.790148496 ; ; c_{2} = 105.246535444 ; ; ; ; (c -257.790148496) (c -105.246535444) = 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 = 248.6 ; ; b = 186.2 ; ; c = 257.79 ; ;

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

p = a+b+c = 248.6+186.2+257.79 = 692.59 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 692.59 }{ 2 } = 346.3 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 346.3 * (346.3-248.6)(346.3-186.2)(346.3-257.79) } ; ; T = sqrt{ 479362871.69 } = 21894.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21894.36 }{ 248.6 } = 176.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21894.36 }{ 186.2 } = 235.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21894.36 }{ 257.79 } = 169.86 ; ;

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{ 248.6**2-186.2**2-257.79**2 }{ 2 * 186.2 * 257.79 } ) = 65° 49'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 186.2**2-248.6**2-257.79**2 }{ 2 * 248.6 * 257.79 } ) = 43° 6' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 257.79**2-248.6**2-186.2**2 }{ 2 * 186.2 * 248.6 } ) = 71° 4'52" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21894.36 }{ 346.3 } = 63.22 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 248.6 }{ 2 * sin 65° 49'8" } = 136.26 ; ;





#2 Obtuse scalene triangle.

Sides: a = 248.6   b = 186.2   c = 105.2476535442

Area: T = 8938.686614155
Perimeter: p = 540.0476535442
Semiperimeter: s = 270.0233267721

Angle ∠ A = α = 114.1811155777° = 114°10'52″ = 1.9932837112 rad
Angle ∠ B = β = 43.1° = 43°6' = 0.75222369076 rad
Angle ∠ C = γ = 22.71988442228° = 22°43'8″ = 0.39765186339 rad

Height: ha = 71.9122197438
Height: hb = 96.01216663969
Height: hc = 169.8621860137

Median: ma = 86.15876845745
Median: mb = 166.6498692198
Median: mc = 213.2329903377

Inradius: r = 33.10333922261
Circumradius: R = 136.2565778556

Vertex coordinates: A[105.2476535442; 0] B[0; 0] C[181.5188341968; 169.8621860137]
Centroid: CG[95.58882924699; 56.62106200457]
Coordinates of the circumscribed circle: U[52.62332677208; 125.6843844962]
Coordinates of the inscribed circle: I[83.82332677208; 33.10333922261]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 65.81988442228° = 65°49'8″ = 1.9932837112 rad
∠ B' = β' = 136.9° = 136°54' = 0.75222369076 rad
∠ C' = γ' = 157.2811155777° = 157°16'52″ = 0.39765186339 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 248.6 ; ; b = 186.2 ; ; beta = 43° 6' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 186.2**2 = 248.6**2 + c**2 -2 * 248.6 * c * cos (43° 6') ; ; ; ; c**2 -363.037c +27131.52 =0 ; ; p=1; q=-363.037; r=27131.52 ; ; D = q**2 - 4pr = 363.037**2 - 4 * 1 * 27131.52 = 23269.5538831 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 363.04 ± sqrt{ 23269.55 } }{ 2 } ; ; c_{1,2} = 181.51834197 ± 76.2718065263 ; ; : Nr. 1
c_{1} = 257.790148496 ; ; c_{2} = 105.246535444 ; ; ; ; (c -257.790148496) (c -105.246535444) = 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 = 248.6 ; ; b = 186.2 ; ; c = 105.25 ; ;

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

p = a+b+c = 248.6+186.2+105.25 = 540.05 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 540.05 }{ 2 } = 270.02 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 270.02 * (270.02-248.6)(270.02-186.2)(270.02-105.25) } ; ; T = sqrt{ 79900109.94 } = 8938.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8938.69 }{ 248.6 } = 71.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8938.69 }{ 186.2 } = 96.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8938.69 }{ 105.25 } = 169.86 ; ;

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{ 248.6**2-186.2**2-105.25**2 }{ 2 * 186.2 * 105.25 } ) = 114° 10'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 186.2**2-248.6**2-105.25**2 }{ 2 * 248.6 * 105.25 } ) = 43° 6' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 105.25**2-248.6**2-186.2**2 }{ 2 * 186.2 * 248.6 } ) = 22° 43'8" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8938.69 }{ 270.02 } = 33.1 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 248.6 }{ 2 * sin 114° 10'52" } = 136.26 ; ;




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