Triangle calculator SSA

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Triangle has two solutions with side c=199.9032587408 and with side c=94.54660498791

#1 Obtuse scalene triangle.

Sides: a = 170   b = 100   c = 199.9032587408

Area: T = 8495.865996482
Perimeter: p = 469.9032587408
Semiperimeter: s = 234.9511293704

Angle ∠ A = α = 58.21216693829° = 58°12'42″ = 1.01659852938 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 91.78883306171° = 91°47'18″ = 1.60220085842 rad

Height: ha = 99.95112937038
Height: hb = 169.9177199297
Height: hc = 85

Median: ma = 133.2549848878
Median: mb = 178.6911136395
Median: mc = 97.26111890064

Inradius: r = 36.16600901655
Circumradius: R = 100

Vertex coordinates: A[199.9032587408; 0] B[0; 0] C[147.2244318643; 85]
Centroid: CG[115.7098968684; 28.33333333333]
Coordinates of the circumscribed circle: U[99.95112937038; -3.12107189772]
Coordinates of the inscribed circle: I[134.9511293704; 36.16600901655]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.7888330617° = 121°47'18″ = 1.01659852938 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 88.21216693829° = 88°12'42″ = 1.60220085842 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 = 170 ; ; b = 100 ; ; c = 199.9 ; ;

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

p = a+b+c = 170+100+199.9 = 469.9 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 469.9 }{ 2 } = 234.95 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 234.95 * (234.95-170)(234.95-100)(234.95-199.9) } ; ; T = sqrt{ 72179636.54 } = 8495.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8495.86 }{ 170 } = 99.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8495.86 }{ 100 } = 169.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8495.86 }{ 199.9 } = 85 ; ;

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{ 170**2-100**2-199.9**2 }{ 2 * 100 * 199.9 } ) = 58° 12'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-170**2-199.9**2 }{ 2 * 170 * 199.9 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 199.9**2-170**2-100**2 }{ 2 * 100 * 170 } ) = 91° 47'18" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8495.86 }{ 234.95 } = 36.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 170 }{ 2 * sin 58° 12'42" } = 100 ; ;





#2 Obtuse scalene triangle.

Sides: a = 170   b = 100   c = 94.54660498791

Area: T = 4018.207711986
Perimeter: p = 364.5466049879
Semiperimeter: s = 182.273302494

Angle ∠ A = α = 121.7888330617° = 121°47'18″ = 2.12656073598 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 28.21216693829° = 28°12'42″ = 0.49223865182 rad

Height: ha = 47.27330249395
Height: hb = 80.36441423972
Height: hc = 85

Median: ma = 47.37659197681
Median: mb = 128.1398510113
Median: mc = 131.2076940034

Inradius: r = 22.04549905914
Circumradius: R = 100

Vertex coordinates: A[94.54660498791; 0] B[0; 0] C[147.2244318643; 85]
Centroid: CG[80.59901228408; 28.33333333333]
Coordinates of the circumscribed circle: U[47.27330249395; 88.12107189772]
Coordinates of the inscribed circle: I[82.27330249395; 22.04549905914]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 58.21216693829° = 58°12'42″ = 2.12656073598 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 151.7888330617° = 151°47'18″ = 0.49223865182 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 170 ; ; b = 100 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 100**2 = 170**2 + c**2 -2 * 100 * c * cos (30° ) ; ; ; ; c**2 -294.449c +18900 =0 ; ; p=1; q=-294.448637287; r=18900 ; ; D = q**2 - 4pr = 294.449**2 - 4 * 1 * 18900 = 11100 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 294.45 ± sqrt{ 11100 } }{ 2 } ; ; c_{1,2} = 147.224318643 ± 52.6782687643 ; ; c_{1} = 199.902587408 ; ;
c_{2} = 94.5460498791 ; ; ; ; (c -199.902587408) (c -94.5460498791) = 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 = 170 ; ; b = 100 ; ; c = 94.55 ; ;

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

p = a+b+c = 170+100+94.55 = 364.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 364.55 }{ 2 } = 182.27 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 182.27 * (182.27-170)(182.27-100)(182.27-94.55) } ; ; T = sqrt{ 16145988.46 } = 4018.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4018.21 }{ 170 } = 47.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4018.21 }{ 100 } = 80.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4018.21 }{ 94.55 } = 85 ; ;

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{ 170**2-100**2-94.55**2 }{ 2 * 100 * 94.55 } ) = 121° 47'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-170**2-94.55**2 }{ 2 * 170 * 94.55 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 94.55**2-170**2-100**2 }{ 2 * 100 * 170 } ) = 28° 12'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4018.21 }{ 182.27 } = 22.04 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 170 }{ 2 * sin 121° 47'18" } = 100 ; ;




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