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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 186.2**2 = 248.6**2 + c**2 -2 * 186.2 * c * cos (43° 6') ; ; ; ; c**2 -363.037c +27131.52 =0 ; ; p=1; q=-363.036683936; 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} = 181.518341968 ± 76.2718065263 ; ;
c_{1} = 257.790148494 ; ; c_{2} = 105.246535442 ; ; ; ; (c -257.790148494) (c -105.246535442) = 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 = 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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