Triangle calculator SSA

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Triangle has two solutions with side c=115.3698860353 and with side c=56.65330689478

#1 Acute scalene triangle.

Sides: a = 105   b = 67   c = 115.3698860353

Area: T = 3474.075513882
Perimeter: p = 287.3698860353
Semiperimeter: s = 143.6844430177

Angle ∠ A = α = 64.01223407832° = 64°44″ = 1.11772261086 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 80.98876592168° = 80°59'16″ = 1.41435013068 rad

Height: ha = 66.17328597871
Height: hb = 103.7043735487
Height: hc = 60.22655258169

Median: ma = 78.37988043387
Median: mb = 105.0966322341
Median: mc = 66.5554537901

Inradius: r = 24.17985079605
Circumradius: R = 58.40554676533

Vertex coordinates: A[115.3698860353; 0] B[0; 0] C[86.01109646503; 60.22655258169]
Centroid: CG[67.12766083344; 20.07551752723]
Coordinates of the circumscribed circle: U[57.68444301765; 9.14990527935]
Coordinates of the inscribed circle: I[76.68444301765; 24.17985079605]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.9887659217° = 115°59'16″ = 1.11772261086 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 99.01223407832° = 99°44″ = 1.41435013068 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 105 ; ; b = 67 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 67**2 = 105**2 + c**2 -2 * 105 * c * cos (35° ) ; ; ; ; c**2 -172.022c +6536 =0 ; ; p=1; q=-172.022; r=6536 ; ; D = q**2 - 4pr = 172.022**2 - 4 * 1 * 6536 = 3447.54416033 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 172.02 ± sqrt{ 3447.54 } }{ 2 } ; ; c_{1,2} = 86.01096465 ± 29.3578957026 ; ; c_{1} = 115.368860353 ; ;
c_{2} = 56.6530689474 ; ; ; ; text{ Factored form: } ; ; (c -115.368860353) (c -56.6530689474) = 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 = 105 ; ; b = 67 ; ; c = 115.37 ; ;

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

p = a+b+c = 105+67+115.37 = 287.37 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 287.37 }{ 2 } = 143.68 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 143.68 * (143.68-105)(143.68-67)(143.68-115.37) } ; ; T = sqrt{ 12069198.07 } = 3474.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3474.08 }{ 105 } = 66.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3474.08 }{ 67 } = 103.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3474.08 }{ 115.37 } = 60.23 ; ;

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{ 67**2+115.37**2-105**2 }{ 2 * 67 * 115.37 } ) = 64° 44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 105**2+115.37**2-67**2 }{ 2 * 105 * 115.37 } ) = 35° ; ; gamma = 180° - alpha - beta = 180° - 64° 44" - 35° = 80° 59'16" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3474.08 }{ 143.68 } = 24.18 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 105 }{ 2 * sin 64° 44" } = 58.41 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 115.37**2 - 105**2 } }{ 2 } = 78.379 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 115.37**2+2 * 105**2 - 67**2 } }{ 2 } = 105.096 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 105**2 - 115.37**2 } }{ 2 } = 66.555 ; ;





#2 Obtuse scalene triangle.

Sides: a = 105   b = 67   c = 56.65330689478

Area: T = 1705.988043326
Perimeter: p = 228.6533068948
Semiperimeter: s = 114.3276534474

Angle ∠ A = α = 115.9887659217° = 115°59'16″ = 2.0244366545 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 29.01223407832° = 29°44″ = 0.50663608704 rad

Height: ha = 32.49548653954
Height: hb = 50.92547890525
Height: hc = 60.22655258169

Median: ma = 33.06110815098
Median: mb = 77.42876120683
Median: mc = 83.39442890413

Inradius: r = 14.92219990015
Circumradius: R = 58.40554676533

Vertex coordinates: A[56.65330689478; 0] B[0; 0] C[86.01109646503; 60.22655258169]
Centroid: CG[47.5554677866; 20.07551752723]
Coordinates of the circumscribed circle: U[28.32765344739; 51.07664730233]
Coordinates of the inscribed circle: I[47.32765344739; 14.92219990015]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 64.01223407832° = 64°44″ = 2.0244366545 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 150.9887659217° = 150°59'16″ = 0.50663608704 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 105 ; ; b = 67 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 67**2 = 105**2 + c**2 -2 * 105 * c * cos (35° ) ; ; ; ; c**2 -172.022c +6536 =0 ; ; p=1; q=-172.022; r=6536 ; ; D = q**2 - 4pr = 172.022**2 - 4 * 1 * 6536 = 3447.54416033 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 172.02 ± sqrt{ 3447.54 } }{ 2 } ; ; c_{1,2} = 86.01096465 ± 29.3578957026 ; ; c_{1} = 115.368860353 ; ; : Nr. 1
c_{2} = 56.6530689474 ; ; ; ; text{ Factored form: } ; ; (c -115.368860353) (c -56.6530689474) = 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 = 105 ; ; b = 67 ; ; c = 56.65 ; ;

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

p = a+b+c = 105+67+56.65 = 228.65 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 228.65 }{ 2 } = 114.33 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 114.33 * (114.33-105)(114.33-67)(114.33-56.65) } ; ; T = sqrt{ 2910369.24 } = 1705.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1705.98 }{ 105 } = 32.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1705.98 }{ 67 } = 50.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1705.98 }{ 56.65 } = 60.23 ; ;

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{ 67**2+56.65**2-105**2 }{ 2 * 67 * 56.65 } ) = 115° 59'16" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 105**2+56.65**2-67**2 }{ 2 * 105 * 56.65 } ) = 35° ; ; gamma = 180° - alpha - beta = 180° - 115° 59'16" - 35° = 29° 44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1705.98 }{ 114.33 } = 14.92 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 105 }{ 2 * sin 115° 59'16" } = 58.41 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 56.65**2 - 105**2 } }{ 2 } = 33.061 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 56.65**2+2 * 105**2 - 67**2 } }{ 2 } = 77.428 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 105**2 - 56.65**2 } }{ 2 } = 83.394 ; ;
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