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?

1. Use Law of Cosines

a = 250 ; ; b = 200 ; ; beta = 48° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 200**2 = 250**2 + c**2 -2 * 250 * c * cos (48° ) ; ; ; ; c**2 -334.565c +22500 =0 ; ; p=1; q=-334.565; 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} = 241.333212175 ; ; c_{2} = 93.2320910055 ; ; ; ; text{ Factored form: } ; ; (c -241.333212175) (c -93.2320910055) = 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 = 241.33 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 200**2+241.33**2-250**2 }{ 2 * 200 * 241.33 } ) = 68° 16'8" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 250**2+241.33**2-200**2 }{ 2 * 250 * 241.33 } ) = 48° ; ; gamma = 180° - alpha - beta = 180° - 68° 16'8" - 48° = 63° 43'52" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 241.33**2 - 250**2 } }{ 2 } = 183.019 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 241.33**2+2 * 250**2 - 200**2 } }{ 2 } = 224.435 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 250**2 - 241.33**2 } }{ 2 } = 191.545 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 250 ; ; b = 200 ; ; beta = 48° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 200**2 = 250**2 + c**2 -2 * 250 * c * cos (48° ) ; ; ; ; c**2 -334.565c +22500 =0 ; ; p=1; q=-334.565; 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} = 241.333212175 ; ; c_{2} = 93.2320910055 ; ; ; ; text{ Factored form: } ; ; (c -241.333212175) (c -93.2320910055) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 200**2+93.23**2-250**2 }{ 2 * 200 * 93.23 } ) = 111° 43'52" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 250**2+93.23**2-200**2 }{ 2 * 250 * 93.23 } ) = 48° ; ; gamma = 180° - alpha - beta = 180° - 111° 43'52" - 48° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 93.23**2 - 250**2 } }{ 2 } = 93.387 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 93.23**2+2 * 250**2 - 200**2 } }{ 2 } = 159.988 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 250**2 - 93.23**2 } }{ 2 } = 221.533 ; ;
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