Triangle calculator SSA

Please enter two sides and a non-included angle
°


Obtuse scalene triangle.

Sides: a = 36   b = 45   c = 14.45499614792

Area: T = 225.2532607044
Perimeter: p = 95.45499614792
Semiperimeter: s = 47.72549807396

Angle ∠ A = α = 43.8543778612° = 43°51'14″ = 0.76553928262 rad
Angle ∠ B = β = 120° = 2.09443951024 rad
Angle ∠ C = γ = 16.1466221388° = 16°8'46″ = 0.2821804725 rad

Height: ha = 12.51440337247
Height: hb = 10.01112269797
Height: hc = 31.17769145362

Median: ma = 28.15884923846
Median: mb = 15.68991903352
Median: mc = 40.10436114747

Inradius: r = 4.72198050906
Circumradius: R = 25.98107621135

Vertex coordinates: A[14.45499614792; 0] B[0; 0] C[-18; 31.17769145362]
Centroid: CG[-1.18333461736; 10.39223048454]
Coordinates of the circumscribed circle: U[7.22549807396; 24.95659542657]
Coordinates of the inscribed circle: I[2.72549807396; 4.72198050906]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.1466221388° = 136°8'46″ = 0.76553928262 rad
∠ B' = β' = 60° = 2.09443951024 rad
∠ C' = γ' = 163.8543778612° = 163°51'14″ = 0.2821804725 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 36 ; ; b = 45 ; ; beta = 120° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 45**2 = 36**2 + c**2 -2 * 45 * c * cos (120° ) ; ; ; ; c**2 +36c -729 =0 ; ; p=1; q=36; r=-729 ; ; D = q**2 - 4pr = 36**2 - 4 * 1 * (-729) = 4212 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ -36 ± sqrt{ 4212 } }{ 2 } = -18 ± sqrt{ 1053 } ; ; c_{1,2} = -18 ± 32.4499614792 ; ; c_{1} = 14.4499614792 ; ;
c_{2} = -50.4499614792 ; ; ; ; (c -14.4499614792) (c +50.4499614792) = 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 = 36 ; ; b = 45 ; ; c = 14.45 ; ;

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

p = a+b+c = 36+45+14.45 = 95.45 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 95.45 }{ 2 } = 47.72 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 47.72 * (47.72-36)(47.72-45)(47.72-14.45) } ; ; T = sqrt{ 50738.74 } = 225.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 225.25 }{ 36 } = 12.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 225.25 }{ 45 } = 10.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 225.25 }{ 14.45 } = 31.18 ; ;

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{ 36**2-45**2-14.45**2 }{ 2 * 45 * 14.45 } ) = 43° 51'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-36**2-14.45**2 }{ 2 * 36 * 14.45 } ) = 120° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.45**2-36**2-45**2 }{ 2 * 45 * 36 } ) = 16° 8'46" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 225.25 }{ 47.72 } = 4.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 36 }{ 2 * sin 43° 51'14" } = 25.98 ; ;




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