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?

1. Use Law of Cosines

a = 8.11 ; ; b = 6.6 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6.6**2 = 8.11**2 + c**2 -2 * 8.11 * c * cos (24° ) ; ; ; ; c**2 -14.818c +22.212 =0 ; ; p=1; q=-14.818; r=22.212 ; ; 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} = 7.40885366 ± 5.71655600666 ; ; c_{1} = 13.1254096667 ; ;
c_{2} = 1.69229765334 ; ; ; ; text{ Factored form: } ; ; (c -13.1254096667) (c -1.69229765334) = 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 = 13.13 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.65 }{ 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 ; ;

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

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.6**2+2 * 13.13**2 - 8.11**2 } }{ 2 } = 9.564 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.13**2+2 * 8.11**2 - 6.6**2 } }{ 2 } = 10.399 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.6**2+2 * 8.11**2 - 13.13**2 } }{ 2 } = 3.405 ; ;







#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 - 2ac cos beta ; ; 6.6**2 = 8.11**2 + c**2 -2 * 8.11 * c * cos (24° ) ; ; ; ; c**2 -14.818c +22.212 =0 ; ; p=1; q=-14.818; r=22.212 ; ; 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} = 7.40885366 ± 5.71655600666 ; ; c_{1} = 13.1254096667 ; ; : Nr. 1
c_{2} = 1.69229765334 ; ; ; ; text{ Factored form: } ; ; (c -13.1254096667) (c -1.69229765334) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 6.6**2+1.69**2-8.11**2 }{ 2 * 6.6 * 1.69 } ) = 150° 49" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.11**2+1.69**2-6.6**2 }{ 2 * 8.11 * 1.69 } ) = 24° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 8.11**2+6.6**2-1.69**2 }{ 2 * 8.11 * 6.6 } ) = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.6**2+2 * 1.69**2 - 8.11**2 } }{ 2 } = 2.602 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.69**2+2 * 8.11**2 - 6.6**2 } }{ 2 } = 4.84 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.6**2+2 * 8.11**2 - 1.69**2 } }{ 2 } = 7.345 ; ;
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