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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 4.5**2 = 6.8**2 + c**2 -2 * 4.5 * c * cos (15° 30') ; ; ; ; c**2 -13.105c +25.99 =0 ; ; p=1; q=-13.1053741636; 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} = 6.55268708182 ± 4.11675940422 ; ;
c_{1} = 10.669446486 ; ; c_{2} = 2.4359276776 ; ; ; ; (c -10.669446486) (c -2.4359276776) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 6.8**2-4.5**2-2.44**2 }{ 2 * 4.5 * 2.44 } ) = 156° 10'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.5**2-6.8**2-2.44**2 }{ 2 * 6.8 * 2.44 } ) = 15° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.44**2-6.8**2-4.5**2 }{ 2 * 4.5 * 6.8 } ) = 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 ; ;




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