Triangle calculator

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

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

Obtuse scalene triangle.

Sides: a = 10.2   b = 15.55334998432   c = 7.79656230659

Area: T = 34.77328483704
Perimeter: p = 33.5499122909
Semiperimeter: s = 16.77545614545

Angle ∠ A = α = 35° = 0.61108652382 rad
Angle ∠ B = β = 119° = 2.07769418099 rad
Angle ∠ C = γ = 26° = 0.45437856055 rad

Height: ha = 6.81882055628
Height: hb = 4.47113856972
Height: hc = 8.92111210128

Median: ma = 11.19551573539
Median: mb = 4.68327374632
Median: mc = 12.56111601351

Inradius: r = 2.0732951264
Circumradius: R = 8.89215786577

Vertex coordinates: A[7.79656230659; 0] B[0; 0] C[-4.94550581265; 8.92111210128]
Centroid: CG[0.95501883131; 2.97437070043]
Coordinates of the circumscribed circle: U[3.89878115329; 7.99216979597]
Coordinates of the inscribed circle: I[1.22110616114; 2.0732951264]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145° = 0.61108652382 rad
∠ B' = β' = 61° = 2.07769418099 rad
∠ C' = γ' = 154° = 0.45437856055 rad

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How did we calculate this triangle?

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

a = 10.2 ; ; alpha = 35° ; ; gamma = 26° ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 35 ° - 26 ° = 119 ° ; ;

3. From angle β, angle α and side a we calculate 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 = 10.2 * fraction{ sin 119° }{ sin 35° } = 15.55 ; ;

4. From angle γ, angle α and side a we calculate 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 = 10.2 * fraction{ sin 26° }{ sin 35° } = 7.8 ; ;


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 = 10.2 ; ; b = 15.55 ; ; c = 7.8 ; ;

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

p = a+b+c = 10.2+15.55+7.8 = 33.55 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.55 }{ 2 } = 16.77 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.77 * (16.77-10.2)(16.77-15.55)(16.77-7.8) } ; ; T = sqrt{ 1209.15 } = 34.77 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34.77 }{ 10.2 } = 6.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34.77 }{ 15.55 } = 4.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34.77 }{ 7.8 } = 8.92 ; ;

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{ 15.55**2+7.8**2-10.2**2 }{ 2 * 15.55 * 7.8 } ) = 35° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.2**2+7.8**2-15.55**2 }{ 2 * 10.2 * 7.8 } ) = 119° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 10.2**2+15.55**2-7.8**2 }{ 2 * 10.2 * 15.55 } ) = 26° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34.77 }{ 16.77 } = 2.07 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.2 }{ 2 * sin 35° } = 8.89 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.55**2+2 * 7.8**2 - 10.2**2 } }{ 2 } = 11.195 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.8**2+2 * 10.2**2 - 15.55**2 } }{ 2 } = 4.683 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.55**2+2 * 10.2**2 - 7.8**2 } }{ 2 } = 12.561 ; ;
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