Triangle calculator SSA

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Triangle has two solutions with side c=191.4211356237 and with side c=91.42113562373

#1 Acute scalene triangle.

Sides: a = 200   b = 150   c = 191.4211356237

Area: T = 13535.53439059
Perimeter: p = 541.4211356237
Semiperimeter: s = 270.7110678119

Angle ∠ A = α = 70.52987793655° = 70°31'44″ = 1.23109594173 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 64.47112206345° = 64°28'16″ = 1.12552350729 rad

Height: ha = 135.3555339059
Height: hb = 180.4743785412
Height: hc = 141.4211356237

Median: ma = 139.8976632597
Median: mb = 180.8210540348
Median: mc = 148.6255253891

Inradius: r = 50
Circumradius: R = 106.0666017178

Vertex coordinates: A[191.4211356237; 0] B[0; 0] C[141.4211356237; 141.4211356237]
Centroid: CG[110.9487570825; 47.14404520791]
Coordinates of the circumscribed circle: U[95.71106781187; 45.71106781187]
Coordinates of the inscribed circle: I[120.7110678119; 50]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 109.4711220634° = 109°28'16″ = 1.23109594173 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 115.5298779366° = 115°31'44″ = 1.12552350729 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 = 200 ; ; b = 150 ; ; c = 191.42 ; ;

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

p = a+b+c = 200+150+191.42 = 541.42 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 541.42 }{ 2 } = 270.71 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 270.71 * (270.71-200)(270.71-150)(270.71-191.42) } ; ; T = sqrt{ 183210678.12 } = 13535.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13535.53 }{ 200 } = 135.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13535.53 }{ 150 } = 180.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13535.53 }{ 191.42 } = 141.42 ; ;

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{ 200**2-150**2-191.42**2 }{ 2 * 150 * 191.42 } ) = 70° 31'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 150**2-200**2-191.42**2 }{ 2 * 200 * 191.42 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 191.42**2-200**2-150**2 }{ 2 * 150 * 200 } ) = 64° 28'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13535.53 }{ 270.71 } = 50 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 200 }{ 2 * sin 70° 31'44" } = 106.07 ; ;





#2 Obtuse scalene triangle.

Sides: a = 200   b = 150   c = 91.42113562373

Area: T = 6464.466609407
Perimeter: p = 441.4211356237
Semiperimeter: s = 220.7110678119

Angle ∠ A = α = 109.4711220634° = 109°28'16″ = 1.91106332362 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 25.52987793655° = 25°31'44″ = 0.44655612539 rad

Height: ha = 64.64546609407
Height: hb = 86.19328812542
Height: hc = 141.4211356237

Median: ma = 73.68112879104
Median: mb = 136.2132819471
Median: mc = 170.7654556937

Inradius: r = 29.28993218813
Circumradius: R = 106.0666017178

Vertex coordinates: A[91.42113562373; 0] B[0; 0] C[141.4211356237; 141.4211356237]
Centroid: CG[77.61442374915; 47.14404520791]
Coordinates of the circumscribed circle: U[45.71106781187; 95.71106781187]
Coordinates of the inscribed circle: I[70.71106781187; 29.28993218813]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 70.52987793655° = 70°31'44″ = 1.91106332362 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 154.4711220634° = 154°28'16″ = 0.44655612539 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 200 ; ; b = 150 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 150**2 = 200**2 + c**2 -2 * 150 * c * cos (45° ) ; ; ; ; c**2 -282.843c +17500 =0 ; ; p=1; q=-282.842712475; r=17500 ; ; D = q**2 - 4pr = 282.843**2 - 4 * 1 * 17500 = 10000 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 282.84 ± sqrt{ 10000 } }{ 2 } ; ; c_{1,2} = fraction{ 282.84 ± 100 }{ 2 } ; ; c_{1,2} = 141.421356237 ± 50 ; ;
c_{1} = 191.421356237 ; ; c_{2} = 91.4213562373 ; ; ; ; (c -191.421356237) (c -91.4213562373) = 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 = 200 ; ; b = 150 ; ; c = 91.42 ; ;

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

p = a+b+c = 200+150+91.42 = 441.42 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 441.42 }{ 2 } = 220.71 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 220.71 * (220.71-200)(220.71-150)(220.71-91.42) } ; ; T = sqrt{ 41789321.88 } = 6464.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6464.47 }{ 200 } = 64.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6464.47 }{ 150 } = 86.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6464.47 }{ 91.42 } = 141.42 ; ;

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{ 200**2-150**2-91.42**2 }{ 2 * 150 * 91.42 } ) = 109° 28'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 150**2-200**2-91.42**2 }{ 2 * 200 * 91.42 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 91.42**2-200**2-150**2 }{ 2 * 150 * 200 } ) = 25° 31'44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6464.47 }{ 220.71 } = 29.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 200 }{ 2 * sin 109° 28'16" } = 106.07 ; ;




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