Triangle calculator SSA

Please enter two sides and a non-included angle
°


Acute scalene triangle.

Sides: a = 42   b = 45   c = 40.59552082572

Area: T = 778.7976930214
Perimeter: p = 127.5955208257
Semiperimeter: s = 63.79876041286

Angle ∠ A = α = 58.55002481311° = 58°30'1″ = 1.02110219431 rad
Angle ∠ B = β = 66° = 1.15219173063 rad
Angle ∠ C = γ = 55.54997518689° = 55°29'59″ = 0.96986534042 rad

Height: ha = 37.08655681054
Height: hb = 34.61331968984
Height: hc = 38.3698909221

Median: ma = 37.35661971662
Median: mb = 34.6377197732
Median: mc = 38.50333409802

Inradius: r = 12.2077306855
Circumradius: R = 24.62993162664

Vertex coordinates: A[40.59552082572; 0] B[0; 0] C[17.08329390092; 38.3698909221]
Centroid: CG[19.22660490888; 12.7989636407]
Coordinates of the circumscribed circle: U[20.29876041286; 13.95502862475]
Coordinates of the inscribed circle: I[18.79876041286; 12.2077306855]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.5499751869° = 121°29'59″ = 1.02110219431 rad
∠ B' = β' = 114° = 1.15219173063 rad
∠ C' = γ' = 124.5500248131° = 124°30'1″ = 0.96986534042 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 42 ; ; b = 45 ; ; beta = 66° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 45**2 = 42**2 + c**2 -2 * 42 * c * cos (66° ) ; ; ; ; c**2 -34.166c -261 =0 ; ; p=1; q=-34.166; r=-261 ; ; D = q**2 - 4pr = 34.166**2 - 4 * 1 * (-261) = 2211.30722077 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 34.17 ± sqrt{ 2211.31 } }{ 2 } ; ; c_{1,2} = 17.08293901 ± 23.512269248 ; ; c_{1} = 40.595208258 ; ; c_{2} = -6.42933023802 ; ; ; ; text{ Factored form: } ; ; (c -40.595208258) (c +6.42933023802) = 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 = 42 ; ; b = 45 ; ; c = 40.6 ; ;

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

p = a+b+c = 42+45+40.6 = 127.6 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 127.6 }{ 2 } = 63.8 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.8 * (63.8-42)(63.8-45)(63.8-40.6) } ; ; T = sqrt{ 606524.66 } = 778.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 778.8 }{ 42 } = 37.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 778.8 }{ 45 } = 34.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 778.8 }{ 40.6 } = 38.37 ; ;

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{ 45**2+40.6**2-42**2 }{ 2 * 45 * 40.6 } ) = 58° 30'1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 42**2+40.6**2-45**2 }{ 2 * 42 * 40.6 } ) = 66° ; ; gamma = 180° - alpha - beta = 180° - 58° 30'1" - 66° = 55° 29'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 778.8 }{ 63.8 } = 12.21 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 42 }{ 2 * sin 58° 30'1" } = 24.63 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 40.6**2 - 42**2 } }{ 2 } = 37.356 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 40.6**2+2 * 42**2 - 45**2 } }{ 2 } = 34.637 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 42**2 - 40.6**2 } }{ 2 } = 38.503 ; ;
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