Triangle calculator SSA

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Triangle has two solutions with side c=50.95440389941 and with side c=2.66990720242

#1 Obtuse scalene triangle.

Sides: a = 35   b = 33   c = 50.95440389941

Area: T = 573.1710936256
Perimeter: p = 118.9544038994
Semiperimeter: s = 59.47770194971

Angle ∠ A = α = 42.98801101602° = 42°58'48″ = 0.75501444352 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 97.02198898398° = 97°1'12″ = 1.69333165176 rad

Height: ha = 32.75326249289
Height: hb = 34.73876325003
Height: hc = 22.4987566339

Median: ma = 39.19770285214
Median: mb = 40.47772410733
Median: mc = 22.53771133366

Inradius: r = 9.6376846989
Circumradius: R = 25.66994431432

Vertex coordinates: A[50.95440389941; 0] B[0; 0] C[26.81215555092; 22.4987566339]
Centroid: CG[25.92218648344; 7.49991887797]
Coordinates of the circumscribed circle: U[25.47770194971; -3.13771625442]
Coordinates of the inscribed circle: I[26.47770194971; 9.6376846989]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.021988984° = 137°1'12″ = 0.75501444352 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 82.98801101602° = 82°58'48″ = 1.69333165176 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 33 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 33**2 = 35**2 + c**2 -2 * 35 * c * cos (40° ) ; ; ; ; c**2 -53.623c +136 =0 ; ; p=1; q=-53.623; r=136 ; ; D = q**2 - 4pr = 53.623**2 - 4 * 1 * 136 = 2331.43803528 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 53.62 ± sqrt{ 2331.44 } }{ 2 } ; ; c_{1,2} = 26.81155551 ± 24.1424834849 ; ; c_{1} = 50.9540389949 ; ;
c_{2} = 2.66907202505 ; ; ; ; text{ Factored form: } ; ; (c -50.9540389949) (c -2.66907202505) = 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 = 33 ; ; c = 50.95 ; ;

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

p = a+b+c = 35+33+50.95 = 118.95 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 118.95 }{ 2 } = 59.48 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 59.48 * (59.48-35)(59.48-33)(59.48-50.95) } ; ; T = sqrt{ 328524.92 } = 573.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 573.17 }{ 35 } = 32.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 573.17 }{ 33 } = 34.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 573.17 }{ 50.95 } = 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{ 33**2+50.95**2-35**2 }{ 2 * 33 * 50.95 } ) = 42° 58'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+50.95**2-33**2 }{ 2 * 35 * 50.95 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 42° 58'48" - 40° = 97° 1'12" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 573.17 }{ 59.48 } = 9.64 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 42° 58'48" } = 25.67 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 50.95**2 - 35**2 } }{ 2 } = 39.197 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 50.95**2+2 * 35**2 - 33**2 } }{ 2 } = 40.477 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 35**2 - 50.95**2 } }{ 2 } = 22.537 ; ;







#2 Obtuse scalene triangle.

Sides: a = 35   b = 33   c = 2.66990720242

Area: T = 30.02438124642
Perimeter: p = 70.66990720242
Semiperimeter: s = 35.33545360121

Angle ∠ A = α = 137.021988984° = 137°1'12″ = 2.39114482184 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 2.98801101602° = 2°58'48″ = 0.05220127344 rad

Height: ha = 1.71656464265
Height: hb = 1.82196249978
Height: hc = 22.4987566339

Median: ma = 15.55503045866
Median: mb = 18.54221674228
Median: mc = 33.98985129659

Inradius: r = 0.85497016192
Circumradius: R = 25.66994431432

Vertex coordinates: A[2.66990720242; 0] B[0; 0] C[26.81215555092; 22.4987566339]
Centroid: CG[9.82768758445; 7.49991887797]
Coordinates of the circumscribed circle: U[1.33545360121; 25.63547288832]
Coordinates of the inscribed circle: I[2.33545360121; 0.85497016192]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.98801101602° = 42°58'48″ = 2.39114482184 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 177.021988984° = 177°1'12″ = 0.05220127344 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 33 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 33**2 = 35**2 + c**2 -2 * 35 * c * cos (40° ) ; ; ; ; c**2 -53.623c +136 =0 ; ; p=1; q=-53.623; r=136 ; ; D = q**2 - 4pr = 53.623**2 - 4 * 1 * 136 = 2331.43803528 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 53.62 ± sqrt{ 2331.44 } }{ 2 } ; ; c_{1,2} = 26.81155551 ± 24.1424834849 ; ; c_{1} = 50.9540389949 ; ; : Nr. 1
c_{2} = 2.66907202505 ; ; ; ; text{ Factored form: } ; ; (c -50.9540389949) (c -2.66907202505) = 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 = 33 ; ; c = 2.67 ; ;

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

p = a+b+c = 35+33+2.67 = 70.67 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70.67 }{ 2 } = 35.33 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.33 * (35.33-35)(35.33-33)(35.33-2.67) } ; ; T = sqrt{ 901.43 } = 30.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 30.02 }{ 35 } = 1.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 30.02 }{ 33 } = 1.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 30.02 }{ 2.67 } = 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{ 33**2+2.67**2-35**2 }{ 2 * 33 * 2.67 } ) = 137° 1'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+2.67**2-33**2 }{ 2 * 35 * 2.67 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 137° 1'12" - 40° = 2° 58'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 30.02 }{ 35.33 } = 0.85 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 137° 1'12" } = 25.67 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 2.67**2 - 35**2 } }{ 2 } = 15.55 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.67**2+2 * 35**2 - 33**2 } }{ 2 } = 18.542 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 35**2 - 2.67**2 } }{ 2 } = 33.989 ; ;
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