Triangle calculator SSA

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Triangle has two solutions with side c=10.6699446486 and with side c=2.43659276776

#1 Obtuse scalene triangle.

Sides: a = 6.8   b = 4.5   c = 10.6699446486

Area: T = 9.69443708788
Perimeter: p = 21.9699446486
Semiperimeter: s = 10.9854723243

Angle ∠ A = α = 23.81876342374° = 23°49'3″ = 0.41656961375 rad
Angle ∠ B = β = 15.5° = 15°30' = 0.27105260341 rad
Angle ∠ C = γ = 140.6822365763° = 140°40'57″ = 2.45553704821 rad

Height: ha = 2.85112855526
Height: hb = 4.30986092795
Height: hc = 1.81772209573

Median: ma = 7.44987276873
Median: mb = 8.65988708363
Median: mc = 2.18876306636

Inradius: r = 0.88325321007
Circumradius: R = 8.41994494556

Vertex coordinates: A[10.6699446486; 0] B[0; 0] C[6.55326870818; 1.81772209573]
Centroid: CG[5.74107111893; 0.60657403191]
Coordinates of the circumscribed circle: U[5.3354723243; -6.51436669439]
Coordinates of the inscribed circle: I[6.4854723243; 0.88325321007]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.1822365763° = 156°10'57″ = 0.41656961375 rad
∠ B' = β' = 164.5° = 164°30' = 0.27105260341 rad
∠ C' = γ' = 39.31876342374° = 39°19'3″ = 2.45553704821 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 6.8 ; ; b = 4.5 ; ; beta = 15° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.5**2 = 6.8**2 + c**2 -2 * 6.8 * c * cos (15° 30') ; ; ; ; c**2 -13.105c +25.99 =0 ; ; p=1; q=-13.105; r=25.99 ; ; D = q**2 - 4pr = 13.105**2 - 4 * 1 * 25.99 = 67.7908319689 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 13.11 ± sqrt{ 67.79 } }{ 2 } ; ; c_{1,2} = 6.55268708 ± 4.11675940422 ; ; c_{1} = 10.6694464842 ; ; c_{2} = 2.43592767578 ; ; ; ; text{ Factored form: } ; ; (c -10.6694464842) (c -2.43592767578) = 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 = 6.8 ; ; b = 4.5 ; ; c = 10.67 ; ;

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

p = a+b+c = 6.8+4.5+10.67 = 21.97 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.97 }{ 2 } = 10.98 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.98 * (10.98-6.8)(10.98-4.5)(10.98-10.67) } ; ; T = sqrt{ 93.98 } = 9.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9.69 }{ 6.8 } = 2.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9.69 }{ 4.5 } = 4.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9.69 }{ 10.67 } = 1.82 ; ;

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{ 4.5**2+10.67**2-6.8**2 }{ 2 * 4.5 * 10.67 } ) = 23° 49'3" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.8**2+10.67**2-4.5**2 }{ 2 * 6.8 * 10.67 } ) = 15° 30' ; ; gamma = 180° - alpha - beta = 180° - 23° 49'3" - 15° 30' = 140° 40'57" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9.69 }{ 10.98 } = 0.88 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.8 }{ 2 * sin 23° 49'3" } = 8.42 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.5**2+2 * 10.67**2 - 6.8**2 } }{ 2 } = 7.449 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.67**2+2 * 6.8**2 - 4.5**2 } }{ 2 } = 8.659 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.5**2+2 * 6.8**2 - 10.67**2 } }{ 2 } = 2.188 ; ;







#2 Obtuse scalene triangle.

Sides: a = 6.8   b = 4.5   c = 2.43659276776

Area: T = 2.21333094131
Perimeter: p = 13.73659276776
Semiperimeter: s = 6.86879638388

Angle ∠ A = α = 156.1822365763° = 156°10'57″ = 2.72658965161 rad
Angle ∠ B = β = 15.5° = 15°30' = 0.27105260341 rad
Angle ∠ C = γ = 8.31876342374° = 8°19'3″ = 0.14551701034 rad

Height: ha = 0.65109733568
Height: hb = 0.98436930725
Height: hc = 1.81772209573

Median: ma = 1.23876880969
Median: mb = 4.5855234108
Median: mc = 5.63657398882

Inradius: r = 0.32222657348
Circumradius: R = 8.41994494556

Vertex coordinates: A[2.43659276776; 0] B[0; 0] C[6.55326870818; 1.81772209573]
Centroid: CG[2.99662049198; 0.60657403191]
Coordinates of the circumscribed circle: U[1.21879638388; 8.33108879013]
Coordinates of the inscribed circle: I[2.36879638388; 0.32222657348]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 23.81876342374° = 23°49'3″ = 2.72658965161 rad
∠ B' = β' = 164.5° = 164°30' = 0.27105260341 rad
∠ C' = γ' = 171.6822365763° = 171°40'57″ = 0.14551701034 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 6.8 ; ; b = 4.5 ; ; beta = 15° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.5**2 = 6.8**2 + c**2 -2 * 6.8 * c * cos (15° 30') ; ; ; ; c**2 -13.105c +25.99 =0 ; ; p=1; q=-13.105; r=25.99 ; ; D = q**2 - 4pr = 13.105**2 - 4 * 1 * 25.99 = 67.7908319689 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 13.11 ± sqrt{ 67.79 } }{ 2 } ; ; c_{1,2} = 6.55268708 ± 4.11675940422 ; ; c_{1} = 10.6694464842 ; ; c_{2} = 2.43592767578 ; ; ; ; text{ Factored form: } ; ; (c -10.6694464842) (c -2.43592767578) = 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 = 6.8 ; ; b = 4.5 ; ; c = 2.44 ; ;

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

p = a+b+c = 6.8+4.5+2.44 = 13.74 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13.74 }{ 2 } = 6.87 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.87 * (6.87-6.8)(6.87-4.5)(6.87-2.44) } ; ; T = sqrt{ 4.9 } = 2.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.21 }{ 6.8 } = 0.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.21 }{ 4.5 } = 0.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.21 }{ 2.44 } = 1.82 ; ;

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{ 4.5**2+2.44**2-6.8**2 }{ 2 * 4.5 * 2.44 } ) = 156° 10'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.8**2+2.44**2-4.5**2 }{ 2 * 6.8 * 2.44 } ) = 15° 30' ; ; gamma = 180° - alpha - beta = 180° - 156° 10'57" - 15° 30' = 8° 19'3" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.21 }{ 6.87 } = 0.32 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.8 }{ 2 * sin 156° 10'57" } = 8.42 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.5**2+2 * 2.44**2 - 6.8**2 } }{ 2 } = 1.238 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.44**2+2 * 6.8**2 - 4.5**2 } }{ 2 } = 4.585 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.5**2+2 * 6.8**2 - 2.44**2 } }{ 2 } = 5.636 ; ;
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