Triangle calculator SSA

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Triangle has two solutions with side c=8.04439907611 and with side c=3.71333433997

#1 Obtuse scalene triangle.

Sides: a = 6.01   b = 2.5   c = 8.04439907611

Area: T = 5.02656813588
Perimeter: p = 16.55439907611
Semiperimeter: s = 8.27769953806

Angle ∠ A = α = 29.98880724774° = 29°59'17″ = 0.52333906011 rad
Angle ∠ B = β = 12° = 0.20994395102 rad
Angle ∠ C = γ = 138.0121927523° = 138°43″ = 2.40987625423 rad

Height: ha = 1.67224397201
Height: hb = 4.0210545087
Height: hc = 1.25495492618

Median: ma = 5.14327491367
Median: mb = 6.98993092422
Median: mc = 2.23879908755

Inradius: r = 0.60771866816
Circumradius: R = 6.01221679309

Vertex coordinates: A[8.04439907611; 0] B[0; 0] C[5.87986670804; 1.25495492618]
Centroid: CG[4.64108859472; 0.41765164206]
Coordinates of the circumscribed circle: U[4.02219953806; -4.46987488616]
Coordinates of the inscribed circle: I[5.77769953806; 0.60771866816]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0121927523° = 150°43″ = 0.52333906011 rad
∠ B' = β' = 168° = 0.20994395102 rad
∠ C' = γ' = 41.98880724774° = 41°59'17″ = 2.40987625423 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 6.01 ; ; b = 2.5 ; ; beta = 12° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.5**2 = 6.01**2 + c**2 -2 * 6.01 * c * cos (12° ) ; ; ; ; c**2 -11.757c +29.87 =0 ; ; p=1; q=-11.757; r=29.87 ; ; D = q**2 - 4pr = 11.757**2 - 4 * 1 * 29.87 = 18.7545065692 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.76 ± sqrt{ 18.75 } }{ 2 } ; ; c_{1,2} = 5.87866708 ± 2.16532368072 ; ; c_{1} = 8.04399076072 ; ;
c_{2} = 3.71334339928 ; ; ; ; text{ Factored form: } ; ; (c -8.04399076072) (c -3.71334339928) = 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 = 6.01 ; ; b = 2.5 ; ; c = 8.04 ; ;

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

p = a+b+c = 6.01+2.5+8.04 = 16.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.55 }{ 2 } = 8.28 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.28 * (8.28-6.01)(8.28-2.5)(8.28-8.04) } ; ; T = sqrt{ 25.26 } = 5.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.03 }{ 6.01 } = 1.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.03 }{ 2.5 } = 4.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.03 }{ 8.04 } = 1.25 ; ;

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{ 2.5**2+8.04**2-6.01**2 }{ 2 * 2.5 * 8.04 } ) = 29° 59'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.01**2+8.04**2-2.5**2 }{ 2 * 6.01 * 8.04 } ) = 12° ; ; gamma = 180° - alpha - beta = 180° - 29° 59'17" - 12° = 138° 43" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.03 }{ 8.28 } = 0.61 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.01 }{ 2 * sin 29° 59'17" } = 6.01 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 8.04**2 - 6.01**2 } }{ 2 } = 5.143 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.04**2+2 * 6.01**2 - 2.5**2 } }{ 2 } = 6.989 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 6.01**2 - 8.04**2 } }{ 2 } = 2.238 ; ;





#2 Obtuse scalene triangle.

Sides: a = 6.01   b = 2.5   c = 3.71333433997

Area: T = 2.3220002752
Perimeter: p = 12.22333433997
Semiperimeter: s = 6.11216716998

Angle ∠ A = α = 150.0121927523° = 150°43″ = 2.61882020525 rad
Angle ∠ B = β = 12° = 0.20994395102 rad
Angle ∠ C = γ = 17.98880724774° = 17°59'17″ = 0.31439510908 rad

Height: ha = 0.77220475048
Height: hb = 1.85660022016
Height: hc = 1.25495492618

Median: ma = 0.99547032733
Median: mb = 4.83765286727
Median: mc = 4.21216291621

Inradius: r = 0.3879601992
Circumradius: R = 6.01221679309

Vertex coordinates: A[3.71333433997; 0] B[0; 0] C[5.87986670804; 1.25495492618]
Centroid: CG[3.19773368267; 0.41765164206]
Coordinates of the circumscribed circle: U[1.85766716998; 5.71882981235]
Coordinates of the inscribed circle: I[3.61216716998; 0.3879601992]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.98880724774° = 29°59'17″ = 2.61882020525 rad
∠ B' = β' = 168° = 0.20994395102 rad
∠ C' = γ' = 162.0121927523° = 162°43″ = 0.31439510908 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 6.01 ; ; b = 2.5 ; ; beta = 12° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.5**2 = 6.01**2 + c**2 -2 * 6.01 * c * cos (12° ) ; ; ; ; c**2 -11.757c +29.87 =0 ; ; p=1; q=-11.757; r=29.87 ; ; D = q**2 - 4pr = 11.757**2 - 4 * 1 * 29.87 = 18.7545065692 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.76 ± sqrt{ 18.75 } }{ 2 } ; ; c_{1,2} = 5.87866708 ± 2.16532368072 ; ; c_{1} = 8.04399076072 ; ; : Nr. 1
c_{2} = 3.71334339928 ; ; ; ; text{ Factored form: } ; ; (c -8.04399076072) (c -3.71334339928) = 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 = 6.01 ; ; b = 2.5 ; ; c = 3.71 ; ;

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

p = a+b+c = 6.01+2.5+3.71 = 12.22 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.22 }{ 2 } = 6.11 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.11 * (6.11-6.01)(6.11-2.5)(6.11-3.71) } ; ; T = sqrt{ 5.38 } = 2.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.32 }{ 6.01 } = 0.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.32 }{ 2.5 } = 1.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.32 }{ 3.71 } = 1.25 ; ;

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{ 2.5**2+3.71**2-6.01**2 }{ 2 * 2.5 * 3.71 } ) = 150° 43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.01**2+3.71**2-2.5**2 }{ 2 * 6.01 * 3.71 } ) = 12° ; ; gamma = 180° - alpha - beta = 180° - 150° 43" - 12° = 17° 59'17" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.32 }{ 6.11 } = 0.38 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.01 }{ 2 * sin 150° 43" } = 6.01 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 3.71**2 - 6.01**2 } }{ 2 } = 0.995 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.71**2+2 * 6.01**2 - 2.5**2 } }{ 2 } = 4.837 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 6.01**2 - 3.71**2 } }{ 2 } = 4.212 ; ;
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