Triangle calculator SSA

Please enter two sides and a non-included angle
°


Obtuse scalene triangle.

Sides: a = 5.6   b = 6.7   c = 10.1587713162

Area: T = 17.51103954795
Perimeter: p = 22.4587713162
Semiperimeter: s = 11.2298856581

Angle ∠ A = α = 30.97695678664° = 30°58'10″ = 0.54105209272 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 111.0330432134° = 111°1'50″ = 1.93878466106 rad

Height: ha = 6.25437126713
Height: hb = 5.22769837252
Height: hc = 3.44877042618

Median: ma = 8.13660044457
Median: mb = 7.48664589988
Median: mc = 3.51114407057

Inradius: r = 1.55994103775
Circumradius: R = 5.44113019724

Vertex coordinates: A[10.1587713162; 0] B[0; 0] C[4.41328602202; 3.44877042618]
Centroid: CG[4.85768577941; 1.14992347539]
Coordinates of the circumscribed circle: U[5.0798856581; -1.95326860946]
Coordinates of the inscribed circle: I[4.5298856581; 1.55994103775]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.0330432134° = 149°1'50″ = 0.54105209272 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 68.97695678664° = 68°58'10″ = 1.93878466106 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 5.6 ; ; b = 6.7 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 6.7**2 = 5.6**2 + c**2 -2 * 6.7 * c * cos (38° ) ; ; ; ; c**2 -8.826c -13.53 =0 ; ; p=1; q=-8.8257204404; r=-13.53 ; ; D = q**2 - 4pr = 8.826**2 - 4 * 1 * (-13.53) = 132.013341292 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 8.83 ± sqrt{ 132.01 } }{ 2 } ; ; c_{1,2} = 4.4128602202 ± 5.74485294181 ; ; c_{1} = 10.157713162 ; ;
c_{2} = -1.33199272161 ; ; ; ; (c -10.157713162) (c +1.33199272161) = 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 = 5.6 ; ; b = 6.7 ; ; c = 10.16 ; ;

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

p = a+b+c = 5.6+6.7+10.16 = 22.46 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.46 }{ 2 } = 11.23 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.23 * (11.23-5.6)(11.23-6.7)(11.23-10.16) } ; ; T = sqrt{ 306.61 } = 17.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.51 }{ 5.6 } = 6.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.51 }{ 6.7 } = 5.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.51 }{ 10.16 } = 3.45 ; ;

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{ 5.6**2-6.7**2-10.16**2 }{ 2 * 6.7 * 10.16 } ) = 30° 58'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.7**2-5.6**2-10.16**2 }{ 2 * 5.6 * 10.16 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.16**2-5.6**2-6.7**2 }{ 2 * 6.7 * 5.6 } ) = 111° 1'50" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.51 }{ 11.23 } = 1.56 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.6 }{ 2 * sin 30° 58'10" } = 5.44 ; ;




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