Triangle calculator SSA

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Triangle has two solutions with side c=11.15985157139 and with side c=1.1654859228

#1 Obtuse scalene triangle.

Sides: a = 6.41   b = 5.3   c = 11.15985157139

Area: T = 9.85876305708
Perimeter: p = 22.86985157139
Semiperimeter: s = 11.4344257857

Angle ∠ A = α = 19.47331559742° = 19°28'23″ = 0.34398706875 rad
Angle ∠ B = β = 16° = 0.27992526803 rad
Angle ∠ C = γ = 144.5276844026° = 144°31'37″ = 2.52224692858 rad

Height: ha = 3.07657037662
Height: hb = 3.72198605927
Height: hc = 1.76768354508

Median: ma = 8.12658360474
Median: mb = 8.70550437373
Median: mc = 1.8660357967

Inradius: r = 0.86221137195
Circumradius: R = 9.61440814881

Vertex coordinates: A[11.15985157139; 0] B[0; 0] C[6.1621687471; 1.76768354508]
Centroid: CG[5.77334010616; 0.58989451503]
Coordinates of the circumscribed circle: U[5.5799257857; -7.83295877686]
Coordinates of the inscribed circle: I[6.1344257857; 0.86221137195]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.5276844026° = 160°31'37″ = 0.34398706875 rad
∠ B' = β' = 164° = 0.27992526803 rad
∠ C' = γ' = 35.47331559742° = 35°28'23″ = 2.52224692858 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 6.41 ; ; b = 5.3 ; ; beta = 16° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.3**2 = 6.41**2 + c**2 -2 * 6.41 * c * cos (16° ) ; ; ; ; c**2 -12.323c +12.998 =0 ; ; p=1; q=-12.323; r=12.998 ; ; D = q**2 - 4pr = 12.323**2 - 4 * 1 * 12.998 = 99.8731699594 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 12.32 ± sqrt{ 99.87 } }{ 2 } ; ; c_{1,2} = 6.16168747 ± 4.99682824298 ; ; c_{1} = 11.158515713 ; ; c_{2} = 1.16485922702 ; ; ; ; text{ Factored form: } ; ; (c -11.158515713) (c -1.16485922702) = 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 = 6.41 ; ; b = 5.3 ; ; c = 11.16 ; ;

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

p = a+b+c = 6.41+5.3+11.16 = 22.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.87 }{ 2 } = 11.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.43 * (11.43-6.41)(11.43-5.3)(11.43-11.16) } ; ; T = sqrt{ 97.17 } = 9.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9.86 }{ 6.41 } = 3.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9.86 }{ 5.3 } = 3.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9.86 }{ 11.16 } = 1.77 ; ;

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{ 5.3**2+11.16**2-6.41**2 }{ 2 * 5.3 * 11.16 } ) = 19° 28'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.41**2+11.16**2-5.3**2 }{ 2 * 6.41 * 11.16 } ) = 16° ; ; gamma = 180° - alpha - beta = 180° - 19° 28'23" - 16° = 144° 31'37" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9.86 }{ 11.43 } = 0.86 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.41 }{ 2 * sin 19° 28'23" } = 9.61 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.3**2+2 * 11.16**2 - 6.41**2 } }{ 2 } = 8.126 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.16**2+2 * 6.41**2 - 5.3**2 } }{ 2 } = 8.705 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.3**2+2 * 6.41**2 - 11.16**2 } }{ 2 } = 1.86 ; ;







#2 Obtuse scalene triangle.

Sides: a = 6.41   b = 5.3   c = 1.1654859228

Area: T = 1.02990572896
Perimeter: p = 12.8754859228
Semiperimeter: s = 6.4377429614

Angle ∠ A = α = 160.5276844026° = 160°31'37″ = 2.80217219661 rad
Angle ∠ B = β = 16° = 0.27992526803 rad
Angle ∠ C = γ = 3.47331559742° = 3°28'23″ = 0.06106180072 rad

Height: ha = 0.32110787175
Height: hb = 0.38883235055
Height: hc = 1.76768354508

Median: ma = 2.11098396883
Median: mb = 3.76882885386
Median: mc = 5.85223350677

Inradius: r = 0.16598553074
Circumradius: R = 9.61440814881

Vertex coordinates: A[1.1654859228; 0] B[0; 0] C[6.1621687471; 1.76768354508]
Centroid: CG[2.4422182233; 0.58989451503]
Coordinates of the circumscribed circle: U[0.5822429614; 9.59664232194]
Coordinates of the inscribed circle: I[1.1377429614; 0.16598553074]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 19.47331559742° = 19°28'23″ = 2.80217219661 rad
∠ B' = β' = 164° = 0.27992526803 rad
∠ C' = γ' = 176.5276844026° = 176°31'37″ = 0.06106180072 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 6.41 ; ; b = 5.3 ; ; beta = 16° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.3**2 = 6.41**2 + c**2 -2 * 6.41 * c * cos (16° ) ; ; ; ; c**2 -12.323c +12.998 =0 ; ; p=1; q=-12.323; r=12.998 ; ; D = q**2 - 4pr = 12.323**2 - 4 * 1 * 12.998 = 99.8731699594 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 12.32 ± sqrt{ 99.87 } }{ 2 } ; ; c_{1,2} = 6.16168747 ± 4.99682824298 ; ; c_{1} = 11.158515713 ; ; c_{2} = 1.16485922702 ; ; ; ; text{ Factored form: } ; ; (c -11.158515713) (c -1.16485922702) = 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 = 6.41 ; ; b = 5.3 ; ; c = 1.16 ; ;

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

p = a+b+c = 6.41+5.3+1.16 = 12.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.87 }{ 2 } = 6.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.44 * (6.44-6.41)(6.44-5.3)(6.44-1.16) } ; ; T = sqrt{ 1.06 } = 1.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.03 }{ 6.41 } = 0.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.03 }{ 5.3 } = 0.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.03 }{ 1.16 } = 1.77 ; ;

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{ 5.3**2+1.16**2-6.41**2 }{ 2 * 5.3 * 1.16 } ) = 160° 31'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.41**2+1.16**2-5.3**2 }{ 2 * 6.41 * 1.16 } ) = 16° ; ; gamma = 180° - alpha - beta = 180° - 160° 31'37" - 16° = 3° 28'23" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.03 }{ 6.44 } = 0.16 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.41 }{ 2 * sin 160° 31'37" } = 9.61 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.3**2+2 * 1.16**2 - 6.41**2 } }{ 2 } = 2.11 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.16**2+2 * 6.41**2 - 5.3**2 } }{ 2 } = 3.768 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.3**2+2 * 6.41**2 - 1.16**2 } }{ 2 } = 5.852 ; ;
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