Triangle calculator - result

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

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

Obtuse scalene triangle.

Sides: a = 3   b = 1.34992228812   c = 2.25774619579

Area: T = 1.43110669729
Perimeter: p = 6.60766848391
Semiperimeter: s = 3.30333424195

Angle ∠ A = α = 110° = 1.92198621772 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 0.95440446486
Height: hb = 2.12113203436
Height: hc = 1.26878547852

Median: ma = 1.09992126442
Median: mb = 2.5687677287
Median: mc = 2.03437570083

Inradius: r = 0.43332178718
Circumradius: R = 1.59662666587

Vertex coordinates: A[2.25774619579; 0] B[0; 0] C[2.71989233611; 1.26878547852]
Centroid: CG[1.65987951063; 0.42326182617]
Coordinates of the circumscribed circle: U[1.1298730979; 1.1298730979]
Coordinates of the inscribed circle: I[1.95441195384; 0.43332178718]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 70° = 1.92198621772 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 135° = 0.78553981634 rad

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How did we calculate this triangle?

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

a = 3 ; ; beta = 25° ; ; gamma = 45° ; ;

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

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 25 ° - 45 ° = 110 ° ; ;

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

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 3 * fraction{ sin 25° }{ sin 110° } = 1.35 ; ;

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

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 3 * fraction{ sin 45° }{ sin 110° } = 2.26 ; ;


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 = 3 ; ; b = 1.35 ; ; c = 2.26 ; ;

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

p = a+b+c = 3+1.35+2.26 = 6.61 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 6.61 }{ 2 } = 3.3 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.3 * (3.3-3)(3.3-1.35)(3.3-2.26) } ; ; T = sqrt{ 2.05 } = 1.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.43 }{ 3 } = 0.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.43 }{ 1.35 } = 2.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.43 }{ 2.26 } = 1.27 ; ;

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{ 1.35**2+2.26**2-3**2 }{ 2 * 1.35 * 2.26 } ) = 110° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3**2+2.26**2-1.35**2 }{ 2 * 3 * 2.26 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 110° - 25° = 45° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.43 }{ 3.3 } = 0.43 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3 }{ 2 * sin 110° } = 1.6 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.35**2+2 * 2.26**2 - 3**2 } }{ 2 } = 1.099 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.26**2+2 * 3**2 - 1.35**2 } }{ 2 } = 2.568 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.35**2+2 * 3**2 - 2.26**2 } }{ 2 } = 2.034 ; ;
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