Triangle calculator SSA

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Triangle has two solutions with side c=8.15876107949 and with side c=1.15884274167

#1 Acute scalene triangle.

Sides: a = 7.1   b = 6.4   c = 8.15876107949

Area: T = 21.85660259162
Perimeter: p = 21.65876107949
Semiperimeter: s = 10.82988053975

Angle ∠ A = α = 56.85114780964° = 56°51'5″ = 0.99222454774 rad
Angle ∠ B = β = 49° = 0.85552113335 rad
Angle ∠ C = γ = 74.14985219036° = 74°8'55″ = 1.29441358427 rad

Height: ha = 6.15766270187
Height: hb = 6.83300080988
Height: hc = 5.35884380196

Median: ma = 6.41548894722
Median: mb = 6.94553802589
Median: mc = 5.39896518004

Inradius: r = 2.01883229003
Circumradius: R = 4.24400415787

Vertex coordinates: A[8.15876107949; 0] B[0; 0] C[4.65880191058; 5.35884380196]
Centroid: CG[4.27218766336; 1.78661460065]
Coordinates of the circumscribed circle: U[4.07988053975; 1.15881446882]
Coordinates of the inscribed circle: I[4.42988053975; 2.01883229003]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.1498521904° = 123°8'55″ = 0.99222454774 rad
∠ B' = β' = 131° = 0.85552113335 rad
∠ C' = γ' = 105.8511478096° = 105°51'5″ = 1.29441358427 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 7.1 ; ; b = 6.4 ; ; beta = 49° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6.4**2 = 7.1**2 + c**2 -2 * 7.1 * c * cos (49° ) ; ; ; ; c**2 -9.316c +9.45 =0 ; ; p=1; q=-9.316; r=9.45 ; ; D = q**2 - 4pr = 9.316**2 - 4 * 1 * 9.45 = 48.9885679612 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 9.32 ± sqrt{ 48.99 } }{ 2 } ; ; c_{1,2} = 4.65801911 ± 3.49959168908 ; ; c_{1} = 8.15761079908 ; ;
c_{2} = 1.15842742092 ; ; ; ; (c -8.15761079908) (c -1.15842742092) = 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 = 7.1 ; ; b = 6.4 ; ; c = 8.16 ; ;

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

p = a+b+c = 7.1+6.4+8.16 = 21.66 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.66 }{ 2 } = 10.83 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.83 * (10.83-7.1)(10.83-6.4)(10.83-8.16) } ; ; T = sqrt{ 477.69 } = 21.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.86 }{ 7.1 } = 6.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.86 }{ 6.4 } = 6.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.86 }{ 8.16 } = 5.36 ; ;

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{ 7.1**2-6.4**2-8.16**2 }{ 2 * 6.4 * 8.16 } ) = 56° 51'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.4**2-7.1**2-8.16**2 }{ 2 * 7.1 * 8.16 } ) = 49° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.16**2-7.1**2-6.4**2 }{ 2 * 6.4 * 7.1 } ) = 74° 8'55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.86 }{ 10.83 } = 2.02 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.1 }{ 2 * sin 56° 51'5" } = 4.24 ; ;





#2 Obtuse scalene triangle.

Sides: a = 7.1   b = 6.4   c = 1.15884274167

Area: T = 3.10436807564
Perimeter: p = 14.65884274167
Semiperimeter: s = 7.32992137084

Angle ∠ A = α = 123.1498521904° = 123°8'55″ = 2.14993471762 rad
Angle ∠ B = β = 49° = 0.85552113335 rad
Angle ∠ C = γ = 7.85114780964° = 7°51'5″ = 0.13770341439 rad

Height: ha = 0.87442762694
Height: hb = 0.97699002364
Height: hc = 5.35884380196

Median: ma = 2.92437778712
Median: mb = 3.95442353294
Median: mc = 6.73442045915

Inradius: r = 0.42334670839
Circumradius: R = 4.24400415787

Vertex coordinates: A[1.15884274167; 0] B[0; 0] C[4.65880191058; 5.35884380196]
Centroid: CG[1.93988155075; 1.78661460065]
Coordinates of the circumscribed circle: U[0.57992137084; 4.22002933313]
Coordinates of the inscribed circle: I[0.92992137084; 0.42334670839]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 56.85114780964° = 56°51'5″ = 2.14993471762 rad
∠ B' = β' = 131° = 0.85552113335 rad
∠ C' = γ' = 172.1498521904° = 172°8'55″ = 0.13770341439 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 7.1 ; ; b = 6.4 ; ; beta = 49° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6.4**2 = 7.1**2 + c**2 -2 * 7.1 * c * cos (49° ) ; ; ; ; c**2 -9.316c +9.45 =0 ; ; p=1; q=-9.316; r=9.45 ; ; D = q**2 - 4pr = 9.316**2 - 4 * 1 * 9.45 = 48.9885679612 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 9.32 ± sqrt{ 48.99 } }{ 2 } ; ; c_{1,2} = 4.65801911 ± 3.49959168908 ; ; c_{1} = 8.15761079908 ; ; : Nr. 1
c_{2} = 1.15842742092 ; ; ; ; (c -8.15761079908) (c -1.15842742092) = 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 = 7.1 ; ; b = 6.4 ; ; c = 1.16 ; ;

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

p = a+b+c = 7.1+6.4+1.16 = 14.66 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.66 }{ 2 } = 7.33 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.33 * (7.33-7.1)(7.33-6.4)(7.33-1.16) } ; ; T = sqrt{ 9.63 } = 3.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.1 }{ 7.1 } = 0.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.1 }{ 6.4 } = 0.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.1 }{ 1.16 } = 5.36 ; ;

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{ 7.1**2-6.4**2-1.16**2 }{ 2 * 6.4 * 1.16 } ) = 123° 8'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.4**2-7.1**2-1.16**2 }{ 2 * 7.1 * 1.16 } ) = 49° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.16**2-7.1**2-6.4**2 }{ 2 * 6.4 * 7.1 } ) = 7° 51'5" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.1 }{ 7.33 } = 0.42 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.1 }{ 2 * sin 123° 8'55" } = 4.24 ; ;




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