Triangle calculator SSA

Please enter two sides and a non-included angle
°


Obtuse scalene triangle.

Sides: a = 8.1   b = 10.6   c = 15.25439095862

Area: T = 40.69327373536
Perimeter: p = 33.95439095862
Semiperimeter: s = 16.97769547931

Angle ∠ A = α = 30.22110985503° = 30°13'16″ = 0.52774576733 rad
Angle ∠ B = β = 41.2° = 41°12' = 0.71990756518 rad
Angle ∠ C = γ = 108.579890145° = 108°34'44″ = 1.89550593285 rad

Height: ha = 10.048758947
Height: hb = 7.67878749724
Height: hc = 5.3355384627

Median: ma = 12.49547340441
Median: mb = 11.00325396537
Median: mc = 5.55110864328

Inradius: r = 2.3976939725
Circumradius: R = 8.04662802594

Vertex coordinates: A[15.25439095862; 0] B[0; 0] C[6.09545607621; 5.3355384627]
Centroid: CG[7.11661567828; 1.77884615423]
Coordinates of the circumscribed circle: U[7.62769547931; -2.56436276243]
Coordinates of the inscribed circle: I[6.37769547931; 2.3976939725]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.779890145° = 149°46'44″ = 0.52774576733 rad
∠ B' = β' = 138.8° = 138°48' = 0.71990756518 rad
∠ C' = γ' = 71.42110985503° = 71°25'16″ = 1.89550593285 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 8.1 ; ; b = 10.6 ; ; beta = 41° 12' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 10.6**2 = 8.1**2 + c**2 -2 * 10.6 * c * cos (41° 12') ; ; ; ; c**2 -12.189c -46.75 =0 ; ; p=1; q=-12.1891215241; r=-46.75 ; ; D = q**2 - 4pr = 12.189**2 - 4 * 1 * (-46.75) = 335.57468353 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 12.19 ± sqrt{ 335.57 } }{ 2 } ; ; c_{1,2} = 6.09456076206 ± 9.15934882415 ; ;
c_{1} = 15.2539095862 ; ; c_{2} = -3.06478806209 ; ; ; ; (c -15.2539095862) (c +3.06478806209) = 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 = 8.1 ; ; b = 10.6 ; ; c = 15.25 ; ;

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

p = a+b+c = 8.1+10.6+15.25 = 33.95 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.95 }{ 2 } = 16.98 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.98 * (16.98-8.1)(16.98-10.6)(16.98-15.25) } ; ; T = sqrt{ 1655.9 } = 40.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40.69 }{ 8.1 } = 10.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40.69 }{ 10.6 } = 7.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40.69 }{ 15.25 } = 5.34 ; ;

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{ 8.1**2-10.6**2-15.25**2 }{ 2 * 10.6 * 15.25 } ) = 30° 13'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.6**2-8.1**2-15.25**2 }{ 2 * 8.1 * 15.25 } ) = 41° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15.25**2-8.1**2-10.6**2 }{ 2 * 10.6 * 8.1 } ) = 108° 34'44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40.69 }{ 16.98 } = 2.4 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.1 }{ 2 * sin 30° 13'16" } = 8.05 ; ;




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