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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 6.4**2 = 7.1**2 + c**2 -2 * 6.4 * c * cos (49° ) ; ; ; ; c**2 -9.316c +9.45 =0 ; ; p=1; q=-9.31603821167; 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} = 4.65801910583 ± 3.49959168908 ; ; c_{1} = 8.15761079492 ; ;
c_{2} = 1.15842741675 ; ; ; ; (c -8.15761079492) (c -1.15842741675) = 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 = 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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