Triangle calculator SSA

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Triangle has two solutions with side c=52.30438976695 and with side c=1.31992133488

#1 Obtuse scalene triangle.

Sides: a = 35   b = 34   c = 52.30438976695

Area: T = 588.3555203805
Perimeter: p = 121.304389767
Semiperimeter: s = 60.65219488348

Angle ∠ A = α = 41.42991285101° = 41°25'45″ = 0.72330746987 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 98.57108714899° = 98°34'15″ = 1.72203862541 rad

Height: ha = 33.62202973603
Height: hb = 34.60991296356
Height: hc = 22.4987566339

Median: ma = 40.49219603837
Median: mb = 41.12660119111
Median: mc = 22.50772337737

Inradius: r = 9.7010516061
Circumradius: R = 26.44773050566

Vertex coordinates: A[52.30438976695; 0] B[0; 0] C[26.81215555092; 22.4987566339]
Centroid: CG[26.37218177262; 7.49991887797]
Coordinates of the circumscribed circle: U[26.15219488348; -3.94215120071]
Coordinates of the inscribed circle: I[26.65219488348; 9.7010516061]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.571087149° = 138°34'15″ = 0.72330746987 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 81.42991285101° = 81°25'45″ = 1.72203862541 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 34 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34**2 = 35**2 + c**2 -2 * 35 * c * cos (40° ) ; ; ; ; c**2 -53.623c +69 =0 ; ; p=1; q=-53.623; r=69 ; ; D = q**2 - 4pr = 53.623**2 - 4 * 1 * 69 = 2599.43803528 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 53.62 ± sqrt{ 2599.44 } }{ 2 } ; ; c_{1,2} = 26.81155551 ± 25.4923421604 ; ; c_{1} = 52.3038976704 ; ;
c_{2} = 1.31921334964 ; ; ; ; text{ Factored form: } ; ; (c -52.3038976704) (c -1.31921334964) = 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 = 35 ; ; b = 34 ; ; c = 52.3 ; ;

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

p = a+b+c = 35+34+52.3 = 121.3 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 121.3 }{ 2 } = 60.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60.65 * (60.65-35)(60.65-34)(60.65-52.3) } ; ; T = sqrt{ 346161.85 } = 588.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 588.36 }{ 35 } = 33.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 588.36 }{ 34 } = 34.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 588.36 }{ 52.3 } = 22.5 ; ;

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{ 34**2+52.3**2-35**2 }{ 2 * 34 * 52.3 } ) = 41° 25'45" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+52.3**2-34**2 }{ 2 * 35 * 52.3 } ) = 40° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 35**2+34**2-52.3**2 }{ 2 * 35 * 34 } ) = 98° 34'15" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 588.36 }{ 60.65 } = 9.7 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 35 }{ 2 * sin 41° 25'45" } = 26.45 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 52.3**2 - 35**2 } }{ 2 } = 40.492 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 52.3**2+2 * 35**2 - 34**2 } }{ 2 } = 41.126 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 35**2 - 52.3**2 } }{ 2 } = 22.507 ; ;







#2 Obtuse scalene triangle.

Sides: a = 35   b = 34   c = 1.31992133488

Area: T = 14.8439544915
Perimeter: p = 70.31992133488
Semiperimeter: s = 35.16596066744

Angle ∠ A = α = 138.571087149° = 138°34'15″ = 2.41985179549 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 1.42991285101° = 1°25'45″ = 0.02549429979 rad

Height: ha = 0.84879739951
Height: hb = 0.87329144068
Height: hc = 22.4987566339

Median: ma = 16.5111213218
Median: mb = 18.01102793407
Median: mc = 34.49773175629

Inradius: r = 0.42220623129
Circumradius: R = 26.44773050566

Vertex coordinates: A[1.31992133488; 0] B[0; 0] C[26.81215555092; 22.4987566339]
Centroid: CG[9.37769229527; 7.49991887797]
Coordinates of the circumscribed circle: U[0.66596066744; 26.43990783461]
Coordinates of the inscribed circle: I[1.16596066744; 0.42220623129]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 41.42991285101° = 41°25'45″ = 2.41985179549 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 178.571087149° = 178°34'15″ = 0.02549429979 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 34 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34**2 = 35**2 + c**2 -2 * 35 * c * cos (40° ) ; ; ; ; c**2 -53.623c +69 =0 ; ; p=1; q=-53.623; r=69 ; ; D = q**2 - 4pr = 53.623**2 - 4 * 1 * 69 = 2599.43803528 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 53.62 ± sqrt{ 2599.44 } }{ 2 } ; ; c_{1,2} = 26.81155551 ± 25.4923421604 ; ; c_{1} = 52.3038976704 ; ; : Nr. 1
c_{2} = 1.31921334964 ; ; ; ; text{ Factored form: } ; ; (c -52.3038976704) (c -1.31921334964) = 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 = 35 ; ; b = 34 ; ; c = 1.32 ; ;

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

p = a+b+c = 35+34+1.32 = 70.32 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70.32 }{ 2 } = 35.16 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.16 * (35.16-35)(35.16-34)(35.16-1.32) } ; ; T = sqrt{ 220.21 } = 14.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.84 }{ 35 } = 0.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.84 }{ 34 } = 0.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.84 }{ 1.32 } = 22.5 ; ;

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{ 34**2+1.32**2-35**2 }{ 2 * 34 * 1.32 } ) = 138° 34'15" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+1.32**2-34**2 }{ 2 * 35 * 1.32 } ) = 40° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 35**2+34**2-1.32**2 }{ 2 * 35 * 34 } ) = 1° 25'45" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.84 }{ 35.16 } = 0.42 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 35 }{ 2 * sin 138° 34'15" } = 26.45 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 1.32**2 - 35**2 } }{ 2 } = 16.511 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.32**2+2 * 35**2 - 34**2 } }{ 2 } = 18.01 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 35**2 - 1.32**2 } }{ 2 } = 34.497 ; ;
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