Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, c and angle α.

Acute scalene triangle.

Sides: a = 7.9   b = 8.71223093636   c = 5.8

Area: T = 22.35495550283
Perimeter: p = 22.41223093636
Semiperimeter: s = 11.20661546818

Angle ∠ A = α = 62.2° = 62°12' = 1.08655947947 rad
Angle ∠ B = β = 77.30106669355° = 77°18'2″ = 1.3499151152 rad
Angle ∠ C = γ = 40.49993330645° = 40°29'58″ = 0.70768467068 rad

Height: ha = 5.6588115197
Height: hb = 5.13105696562
Height: hc = 7.70767431132

Median: ma = 6.25985675057
Median: mb = 5.39897046662
Median: mc = 7.79440469092

Inradius: r = 1.99444000117
Circumradius: R = 4.46553910842

Vertex coordinates: A[5.8; 0] B[0; 0] C[1.73766953063; 7.70767431132]
Centroid: CG[2.51222317688; 2.56989143711]
Coordinates of the circumscribed circle: U[2.9; 3.3965543776]
Coordinates of the inscribed circle: I[2.49438453182; 1.99444000117]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.8° = 117°48' = 1.08655947947 rad
∠ B' = β' = 102.6999333065° = 102°41'58″ = 1.3499151152 rad
∠ C' = γ' = 139.5010666935° = 139°30'2″ = 0.70768467068 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle α.

a = 7.9 ; ; c = 5.8 ; ; alpha = 62.2° ; ;

2. From angle α, side c and side a we calculate side b - by using the law of cosines and quadratic equation:

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 7.9**2 = 5.8**2 + b**2 - 2 * 5.8 * b * cos 62° 12' ; ; ; ; ; ; b**2 -5.41b -28.77 =0 ; ; p=1; q=-5.41; r=-28.77 ; ; D = q**2 - 4pr = 5.41**2 - 4 * 1 * (-28.77) = 144.34902001 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 5.41 ± sqrt{ 144.35 } }{ 2 } ; ;
b_{1,2} = 2.70504251 ± 6.00726684961 ; ; b_{1} = 8.71230936358 ; ; b_{2} = -3.30222433564 ; ; ; ; text{ Factored form: } ; ; (b -8.71230936358) (b +3.30222433564) = 0 ; ; ; ; b > 0 ; ; ; ; b = 8.712 ; ;
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 = 7.9 ; ; b = 8.71 ; ; c = 5.8 ; ;

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

p = a+b+c = 7.9+8.71+5.8 = 22.41 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.41 }{ 2 } = 11.21 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.21 * (11.21-7.9)(11.21-8.71)(11.21-5.8) } ; ; T = sqrt{ 499.5 } = 22.35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.35 }{ 7.9 } = 5.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.35 }{ 8.71 } = 5.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.35 }{ 5.8 } = 7.71 ; ;

7. 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{ 8.71**2+5.8**2-7.9**2 }{ 2 * 8.71 * 5.8 } ) = 62° 12' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.9**2+5.8**2-8.71**2 }{ 2 * 7.9 * 5.8 } ) = 77° 18'2" ; ;
 gamma = 180° - alpha - beta = 180° - 62° 12' - 77° 18'2" = 40° 29'58" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.35 }{ 11.21 } = 1.99 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.9 }{ 2 * sin 62° 12' } = 4.47 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.71**2+2 * 5.8**2 - 7.9**2 } }{ 2 } = 6.259 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.8**2+2 * 7.9**2 - 8.71**2 } }{ 2 } = 5.39 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.71**2+2 * 7.9**2 - 5.8**2 } }{ 2 } = 7.794 ; ;
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