Triangle calculator

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

You have entered side c, angle α and angle β.

Right scalene triangle.

Sides: a = 36.58773301719   b = 17.73878896465   c = 32

Area: T = 283.8066234344
Perimeter: p = 86.32552198184
Semiperimeter: s = 43.16326099092

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 61° = 1.06546508437 rad

Height: ha = 15.51439078479
Height: hb = 32
Height: hc = 17.73878896465

Median: ma = 18.2943665086
Median: mb = 33.20662973286
Median: mc = 23.88879201504

Inradius: r = 6.57552797373
Circumradius: R = 18.2943665086

Vertex coordinates: A[32; 0] B[0; 0] C[32; 17.73878896465]
Centroid: CG[21.33333333333; 5.91326298822]
Coordinates of the circumscribed circle: U[16; 8.86989448232]
Coordinates of the inscribed circle: I[25.42547202627; 6.57552797373]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 119° = 1.06546508437 rad

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How did we calculate this triangle?

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

c = 32 ; ; alpha = 90° ; ; beta = 29° ; ;

2. From angle α and angle β we calculate angle γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 29 ° = 61 ° ; ;

3. From angle α, angle γ and side c we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 32 * fraction{ sin 90° }{ sin 61° } = 36.59 ; ;

4. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 36.59**2+32**2 - 2 * 36.59 * 32 * cos 29° } ; ; b = 17.74 ; ;


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 = 36.59 ; ; b = 17.74 ; ; c = 32 ; ;

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

p = a+b+c = 36.59+17.74+32 = 86.33 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 86.33 }{ 2 } = 43.16 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43.16 * (43.16-36.59)(43.16-17.74)(43.16-32) } ; ; T = sqrt{ 80545.98 } = 283.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 283.81 }{ 36.59 } = 15.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 283.81 }{ 17.74 } = 32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 283.81 }{ 32 } = 17.74 ; ;

9. 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{ 17.74**2+32**2-36.59**2 }{ 2 * 17.74 * 32 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 36.59**2+32**2-17.74**2 }{ 2 * 36.59 * 32 } ) = 29° ; ; gamma = 180° - alpha - beta = 180° - 90° - 29° = 61° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 283.81 }{ 43.16 } = 6.58 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 36.59 }{ 2 * sin 90° } = 18.29 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.74**2+2 * 32**2 - 36.59**2 } }{ 2 } = 18.294 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 36.59**2 - 17.74**2 } }{ 2 } = 33.206 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.74**2+2 * 36.59**2 - 32**2 } }{ 2 } = 23.888 ; ;
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