Triangle calculator SSA

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Triangle has two solutions with side c=302.5440188133 and with side c=25.12106295828

#1 Obtuse scalene triangle.

Sides: a = 200   b = 180   c = 302.5440188133

Area: T = 17352.99222962
Perimeter: p = 682.5440188133
Semiperimeter: s = 341.2770094066

Angle ∠ A = α = 39.59113123899° = 39°35'29″ = 0.69109987564 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 105.409868761° = 105°24'31″ = 1.8439728659 rad

Height: ha = 173.5329922962
Height: hb = 192.8111025513
Height: hc = 114.715528727

Median: ma = 227.9598949633
Median: mb = 240.1365967147
Median: mc = 115.401086066

Inradius: r = 50.84882653415
Circumradius: R = 156.9110211606

Vertex coordinates: A[302.5440188133; 0] B[0; 0] C[163.8330408858; 114.715528727]
Centroid: CG[155.4576865663; 38.23884290901]
Coordinates of the circumscribed circle: U[151.2770094066; -41.69114037585]
Coordinates of the inscribed circle: I[161.2770094066; 50.84882653415]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.409868761° = 140°24'31″ = 0.69109987564 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 74.59113123899° = 74°35'29″ = 1.8439728659 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 = 180 ; ; c = 302.54 ; ;

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

p = a+b+c = 200+180+302.54 = 682.54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 682.54 }{ 2 } = 341.27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 341.27 * (341.27-200)(341.27-180)(341.27-302.54) } ; ; T = sqrt{ 301126341.63 } = 17352.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17352.99 }{ 200 } = 173.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17352.99 }{ 180 } = 192.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17352.99 }{ 302.54 } = 114.72 ; ;

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**2-180**2-302.54**2 }{ 2 * 180 * 302.54 } ) = 39° 35'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 180**2-200**2-302.54**2 }{ 2 * 200 * 302.54 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 302.54**2-200**2-180**2 }{ 2 * 180 * 200 } ) = 105° 24'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17352.99 }{ 341.27 } = 50.85 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 200 }{ 2 * sin 39° 35'29" } = 156.91 ; ;





#2 Obtuse scalene triangle.

Sides: a = 200   b = 180   c = 25.12106295828

Area: T = 1440.86601195
Perimeter: p = 405.1210629583
Semiperimeter: s = 202.5660314791

Angle ∠ A = α = 140.409868761° = 140°24'31″ = 2.45105938972 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 4.59113123899° = 4°35'29″ = 0.08801335182 rad

Height: ha = 14.4098601195
Height: hb = 16.01095568833
Height: hc = 114.715528727

Median: ma = 80.7198789729
Median: mb = 110.5243857222
Median: mc = 189.8487935181

Inradius: r = 7.11332399305
Circumradius: R = 156.9110211606

Vertex coordinates: A[25.12106295828; 0] B[0; 0] C[163.8330408858; 114.715528727]
Centroid: CG[62.98436794802; 38.23884290901]
Coordinates of the circumscribed circle: U[12.56603147914; 156.4076691029]
Coordinates of the inscribed circle: I[22.56603147914; 7.11332399305]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 39.59113123899° = 39°35'29″ = 2.45105938972 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 175.409868761° = 175°24'31″ = 0.08801335182 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 200 ; ; b = 180 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 180**2 = 200**2 + c**2 -2 * 180 * c * cos (35° ) ; ; ; ; c**2 -327.661c +7600 =0 ; ; p=1; q=-327.660817716; r=7600 ; ; D = q**2 - 4pr = 327.661**2 - 4 * 1 * 7600 = 76961.6114661 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 327.66 ± sqrt{ 76961.61 } }{ 2 } ; ; c_{1,2} = 163.830408858 ± 138.709779275 ; ;
c_{1} = 302.540188133 ; ; c_{2} = 25.1206295828 ; ; ; ; (c -302.540188133) (c -25.1206295828) = 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 = 180 ; ; c = 25.12 ; ;

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

p = a+b+c = 200+180+25.12 = 405.12 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 405.12 }{ 2 } = 202.56 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 202.56 * (202.56-200)(202.56-180)(202.56-25.12) } ; ; T = sqrt{ 2076077.88 } = 1440.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1440.86 }{ 200 } = 14.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1440.86 }{ 180 } = 16.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1440.86 }{ 25.12 } = 114.72 ; ;

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**2-180**2-25.12**2 }{ 2 * 180 * 25.12 } ) = 140° 24'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 180**2-200**2-25.12**2 }{ 2 * 200 * 25.12 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25.12**2-200**2-180**2 }{ 2 * 180 * 200 } ) = 4° 35'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1440.86 }{ 202.56 } = 7.11 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 200 }{ 2 * sin 140° 24'31" } = 156.91 ; ;




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