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?

1. Use Law of Cosines

a = 170 ; ; b = 100 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 100**2 = 170**2 + c**2 -2 * 170 * c * cos (30° ) ; ; ; ; c**2 -294.449c +18900 =0 ; ; p=1; q=-294.449; 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} = 147.22431864 ± 52.6782687643 ; ; c_{1} = 199.902587404 ; ;
c_{2} = 94.5460498757 ; ; ; ; (c -199.902587404) (c -94.5460498757) = 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 = 199.9 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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**2-199.9**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" ; ;

7. Inradius

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

8. 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 - 2ac cos beta ; ; 100**2 = 170**2 + c**2 -2 * 170 * c * cos (30° ) ; ; ; ; c**2 -294.449c +18900 =0 ; ; p=1; q=-294.449; 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} = 147.22431864 ± 52.6782687643 ; ; c_{1} = 199.902587404 ; ; : Nr. 1
c_{2} = 94.5460498757 ; ; ; ; (c -199.902587404) (c -94.5460498757) = 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 = 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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