Triangle calculator

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

You have entered side b, angle β and angle γ.

Right scalene triangle.

Sides: a = 8.66602540378   b = 7.5   c = 4.33301270189

Area: T = 16.2387976321
Perimeter: p = 20.49903810568
Semiperimeter: s = 10.24551905284

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 30° = 0.52435987756 rad

Height: ha = 3.75
Height: hb = 4.33301270189
Height: hc = 7.5

Median: ma = 4.33301270189
Median: mb = 5.72882196187
Median: mc = 7.8066247498

Inradius: r = 1.58549364905
Circumradius: R = 4.33301270189

Vertex coordinates: A[4.33301270189; 0] B[0; 0] C[4.33301270189; 7.5]
Centroid: CG[2.88767513459; 2.5]
Coordinates of the circumscribed circle: U[2.16550635095; 3.75]
Coordinates of the inscribed circle: I[2.74551905284; 1.58549364905]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 150° = 0.52435987756 rad

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How did we calculate this triangle?

1. Input data entered: side b, angle β and angle γ.

b = 7.5 ; ; beta = 60° ; ; gamma = 30° ; ;

2. From angle γ, angle β and side b we calculate side c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ b } = fraction{ sin gamma }{ sin beta } ; ; ; ; c = b * fraction{ sin gamma }{ sin beta } ; ; ; ; c = 7.5 * fraction{ sin 30° }{ sin 60° } = 4.33 ; ;

3. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 7.5**2+4.33**2 - 2 * 7.5 * 4.33 * cos 90° } ; ; a = 8.66 ; ;
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 = 8.66 ; ; b = 7.5 ; ; c = 4.33 ; ;

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

p = a+b+c = 8.66+7.5+4.33 = 20.49 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.49 }{ 2 } = 10.25 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.25 * (10.25-8.66)(10.25-7.5)(10.25-4.33) } ; ; T = sqrt{ 263.67 } = 16.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16.24 }{ 8.66 } = 3.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16.24 }{ 7.5 } = 4.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16.24 }{ 4.33 } = 7.5 ; ;

8. 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{ 7.5**2+4.33**2-8.66**2 }{ 2 * 7.5 * 4.33 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.66**2+4.33**2-7.5**2 }{ 2 * 8.66 * 4.33 } ) = 60° ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 60° = 30° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16.24 }{ 10.25 } = 1.58 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.66 }{ 2 * sin 90° } = 4.33 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.5**2+2 * 4.33**2 - 8.66**2 } }{ 2 } = 4.33 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.33**2+2 * 8.66**2 - 7.5**2 } }{ 2 } = 5.728 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.5**2+2 * 8.66**2 - 4.33**2 } }{ 2 } = 7.806 ; ;
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