Triangle calculator SSA

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Triangle has two solutions with side c=13.12554096681 and with side c=1.69222976548

#1 Obtuse scalene triangle.

Sides: a = 8.11   b = 6.6   c = 13.12554096681

Area: T = 21.64879624484
Perimeter: p = 27.83554096681
Semiperimeter: s = 13.91877048341

Angle ∠ A = α = 29.98663096969° = 29°59'11″ = 0.52333598347 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 126.0143690303° = 126°49″ = 2.19993537984 rad

Height: ha = 5.33985850674
Height: hb = 6.56599886207
Height: hc = 3.29986341753

Median: ma = 9.56442649732
Median: mb = 10.39987614396
Median: mc = 3.40554302608

Inradius: r = 1.55554261788
Circumradius: R = 8.11333580074

Vertex coordinates: A[13.12554096681; 0] B[0; 0] C[7.40988536615; 3.29986341753]
Centroid: CG[6.84547544432; 1.10995447251]
Coordinates of the circumscribed circle: U[6.56327048341; -4.77704804178]
Coordinates of the inscribed circle: I[7.31877048341; 1.55554261788]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0143690303° = 150°49″ = 0.52333598347 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 53.98663096969° = 53°59'11″ = 2.19993537984 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 = 8.11 ; ; b = 6.6 ; ; c = 13.13 ; ;

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

p = a+b+c = 8.11+6.6+13.13 = 27.84 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.84 }{ 2 } = 13.92 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.92 * (13.92-8.11)(13.92-6.6)(13.92-13.13) } ; ; T = sqrt{ 468.63 } = 21.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.65 }{ 8.11 } = 5.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.65 }{ 6.6 } = 6.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.65 }{ 13.13 } = 3.3 ; ;

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{ 8.11**2-6.6**2-13.13**2 }{ 2 * 6.6 * 13.13 } ) = 29° 59'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.6**2-8.11**2-13.13**2 }{ 2 * 8.11 * 13.13 } ) = 24° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13.13**2-8.11**2-6.6**2 }{ 2 * 6.6 * 8.11 } ) = 126° 49" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.65 }{ 13.92 } = 1.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.11 }{ 2 * sin 29° 59'11" } = 8.11 ; ;





#2 Obtuse scalene triangle.

Sides: a = 8.11   b = 6.6   c = 1.69222976548

Area: T = 2.79111354395
Perimeter: p = 16.40222976548
Semiperimeter: s = 8.20111488274

Angle ∠ A = α = 150.0143690303° = 150°49″ = 2.61882328189 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 5.98663096969° = 5°59'11″ = 0.10444808143 rad

Height: ha = 0.68883194672
Height: hb = 0.8465798618
Height: hc = 3.29986341753

Median: ma = 2.60217130273
Median: mb = 4.84402464479
Median: mc = 7.34550719644

Inradius: r = 0.34403346895
Circumradius: R = 8.11333580074

Vertex coordinates: A[1.69222976548; 0] B[0; 0] C[7.40988536615; 3.29986341753]
Centroid: CG[3.03437171054; 1.10995447251]
Coordinates of the circumscribed circle: U[0.84661488274; 8.06991145932]
Coordinates of the inscribed circle: I[1.60111488274; 0.34403346895]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.98663096969° = 29°59'11″ = 2.61882328189 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 174.0143690303° = 174°49″ = 0.10444808143 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 8.11 ; ; b = 6.6 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 6.6**2 = 8.11**2 + c**2 -2 * 6.6 * c * cos (24° ) ; ; ; ; c**2 -14.818c +22.212 =0 ; ; p=1; q=-14.817707323; r=22.2121 ; ; D = q**2 - 4pr = 14.818**2 - 4 * 1 * 22.212 = 130.716050309 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 14.82 ± sqrt{ 130.72 } }{ 2 } ; ; c_{1,2} = 7.40885366148 ± 5.71655600666 ; ;
c_{1} = 13.1254096681 ; ; c_{2} = 1.69229765482 ; ; ; ; (c -13.1254096681) (c -1.69229765482) = 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 = 8.11 ; ; b = 6.6 ; ; c = 1.69 ; ;

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

p = a+b+c = 8.11+6.6+1.69 = 16.4 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.4 }{ 2 } = 8.2 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.2 * (8.2-8.11)(8.2-6.6)(8.2-1.69) } ; ; T = sqrt{ 7.79 } = 2.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.79 }{ 8.11 } = 0.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.79 }{ 6.6 } = 0.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.79 }{ 1.69 } = 3.3 ; ;

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{ 8.11**2-6.6**2-1.69**2 }{ 2 * 6.6 * 1.69 } ) = 150° 49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.6**2-8.11**2-1.69**2 }{ 2 * 8.11 * 1.69 } ) = 24° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.69**2-8.11**2-6.6**2 }{ 2 * 6.6 * 8.11 } ) = 5° 59'11" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.79 }{ 8.2 } = 0.34 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.11 }{ 2 * sin 150° 49" } = 8.11 ; ;




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