Triangle calculator SSA

Please enter two sides and a non-included angle
°


Acute scalene triangle.

Sides: a = 85   b = 90   c = 82.45767306549

Area: T = 3176.075502616
Perimeter: p = 257.4576730655
Semiperimeter: s = 128.7288365327

Angle ∠ A = α = 58.86656606444° = 58°51'56″ = 1.02773995946 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 56.13443393556° = 56°8'4″ = 0.98797290452 rad

Height: ha = 74.73111770861
Height: hb = 70.57994450258
Height: hc = 77.03661618981

Median: ma = 75.12219423015
Median: mb = 70.61990924265
Median: mc = 77.21986628506

Inradius: r = 24.67326897998
Circumradius: R = 49.65220063533

Vertex coordinates: A[82.45767306549; 0] B[0; 0] C[35.9232552248; 77.03661618981]
Centroid: CG[39.46597609676; 25.67987206327]
Coordinates of the circumscribed circle: U[41.22883653274; 27.66884590705]
Coordinates of the inscribed circle: I[38.72883653274; 24.67326897998]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.1344339356° = 121°8'4″ = 1.02773995946 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 123.8665660644° = 123°51'56″ = 0.98797290452 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 85 ; ; b = 90 ; ; beta = 65° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 85**2 + c**2 -2 * 90 * c * cos (65° ) ; ; ; ; c**2 -71.845c -875 =0 ; ; p=1; q=-71.8451044959; r=-875 ; ; D = q**2 - 4pr = 71.845**2 - 4 * 1 * (-875) = 8661.71904003 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 71.85 ± sqrt{ 8661.72 } }{ 2 } ; ; c_{1,2} = 35.922552248 ± 46.5341784069 ; ; c_{1} = 82.4567306549 ; ;
c_{2} = -10.611626159 ; ; ; ; (c -82.4567306549) (c +10.611626159) = 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 = 85 ; ; b = 90 ; ; c = 82.46 ; ;

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

p = a+b+c = 85+90+82.46 = 257.46 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 257.46 }{ 2 } = 128.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 128.73 * (128.73-85)(128.73-90)(128.73-82.46) } ; ; T = sqrt{ 10087452.57 } = 3176.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3176.08 }{ 85 } = 74.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3176.08 }{ 90 } = 70.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3176.08 }{ 82.46 } = 77.04 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 85**2-90**2-82.46**2 }{ 2 * 90 * 82.46 } ) = 58° 51'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-85**2-82.46**2 }{ 2 * 85 * 82.46 } ) = 65° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 82.46**2-85**2-90**2 }{ 2 * 90 * 85 } ) = 56° 8'4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3176.08 }{ 128.73 } = 24.67 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 85 }{ 2 * sin 58° 51'56" } = 49.65 ; ;




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