Triangle calculator SSA

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Triangle has two solutions with side c=241.3333212174 and with side c=93.23220910052

#1 Acute scalene triangle.

Sides: a = 250   b = 200   c = 241.3333212174

Area: T = 22418.19109804
Perimeter: p = 691.3333212174
Semiperimeter: s = 345.6676606087

Angle ∠ A = α = 68.26987908647° = 68°16'8″ = 1.19215151769 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 63.73112091353° = 63°43'52″ = 1.11223194357 rad

Height: ha = 179.3465527843
Height: hb = 224.1821909804
Height: hc = 185.7866206369

Median: ma = 183.0198741251
Median: mb = 224.4354533103
Median: mc = 191.5455217052

Inradius: r = 64.85549515215
Circumradius: R = 134.5633272961

Vertex coordinates: A[241.3333212174; 0] B[0; 0] C[167.283265159; 185.7866206369]
Centroid: CG[136.2055287921; 61.92987354564]
Coordinates of the circumscribed circle: U[120.6676606087; 59.55553910682]
Coordinates of the inscribed circle: I[145.6676606087; 64.85549515215]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.7311209135° = 111°43'52″ = 1.19215151769 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 116.2698790865° = 116°16'8″ = 1.11223194357 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 = 250 ; ; b = 200 ; ; c = 241.33 ; ;

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

p = a+b+c = 250+200+241.33 = 691.33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 691.33 }{ 2 } = 345.67 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 345.67 * (345.67-250)(345.67-200)(345.67-241.33) } ; ; T = sqrt{ 502575286.83 } = 22418.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22418.19 }{ 250 } = 179.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22418.19 }{ 200 } = 224.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22418.19 }{ 241.33 } = 185.79 ; ;

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{ 250**2-200**2-241.33**2 }{ 2 * 200 * 241.33 } ) = 68° 16'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 200**2-250**2-241.33**2 }{ 2 * 250 * 241.33 } ) = 48° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 241.33**2-250**2-200**2 }{ 2 * 200 * 250 } ) = 63° 43'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22418.19 }{ 345.67 } = 64.85 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 250 }{ 2 * sin 68° 16'8" } = 134.56 ; ;





#2 Obtuse scalene triangle.

Sides: a = 250   b = 200   c = 93.23220910052

Area: T = 8660.618824987
Perimeter: p = 543.2322091005
Semiperimeter: s = 271.6166045503

Angle ∠ A = α = 111.7311209135° = 111°43'52″ = 1.95500774766 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 20.26987908647° = 20°16'8″ = 0.3543757136 rad

Height: ha = 69.28549459989
Height: hb = 86.60661824987
Height: hc = 185.7866206369

Median: ma = 93.38768909248
Median: mb = 159.9887847653
Median: mc = 221.5333167498

Inradius: r = 31.88655177861
Circumradius: R = 134.5633272961

Vertex coordinates: A[93.23220910052; 0] B[0; 0] C[167.283265159; 185.7866206369]
Centroid: CG[86.83882475316; 61.92987354564]
Coordinates of the circumscribed circle: U[46.61660455026; 126.2310815301]
Coordinates of the inscribed circle: I[71.61660455026; 31.88655177861]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 68.26987908647° = 68°16'8″ = 1.95500774766 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 159.7311209135° = 159°43'52″ = 0.3543757136 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 250 ; ; b = 200 ; ; beta = 48° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 200**2 = 250**2 + c**2 -2 * 200 * c * cos (48° ) ; ; ; ; c**2 -334.565c +22500 =0 ; ; p=1; q=-334.565303179; r=22500 ; ; D = q**2 - 4pr = 334.565**2 - 4 * 1 * 22500 = 21933.9420915 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 334.57 ± sqrt{ 21933.94 } }{ 2 } ; ; c_{1,2} = 167.28265159 ± 74.0505605845 ; ;
c_{1} = 241.333212174 ; ; c_{2} = 93.2320910052 ; ; ; ; (c -241.333212174) (c -93.2320910052) = 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 = 250 ; ; b = 200 ; ; c = 93.23 ; ;

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

p = a+b+c = 250+200+93.23 = 543.23 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 543.23 }{ 2 } = 271.62 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 271.62 * (271.62-250)(271.62-200)(271.62-93.23) } ; ; T = sqrt{ 75006308.47 } = 8660.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8660.62 }{ 250 } = 69.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8660.62 }{ 200 } = 86.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8660.62 }{ 93.23 } = 185.79 ; ;

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{ 250**2-200**2-93.23**2 }{ 2 * 200 * 93.23 } ) = 111° 43'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 200**2-250**2-93.23**2 }{ 2 * 250 * 93.23 } ) = 48° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 93.23**2-250**2-200**2 }{ 2 * 200 * 250 } ) = 20° 16'8" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8660.62 }{ 271.62 } = 31.89 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 250 }{ 2 * sin 111° 43'52" } = 134.56 ; ;




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