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.

Obtuse scalene triangle.

Sides: a = 0.21550320573   b = 0.31112306764   c = 0.5

Area: T = 0.02201381066
Perimeter: p = 1.02662627337
Semiperimeter: s = 0.51331313668

Angle ∠ A = α = 15° = 0.26217993878 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 143° = 2.49658208304 rad

Height: ha = 0.18773032967
Height: hb = 0.12994095226
Height: hc = 0.08105524265

Median: ma = 0.40223339043
Median: mb = 0.35219989479
Median: mc = 0.09551402112

Inradius: r = 0.03992455186
Circumradius: R = 0.41554100353

Vertex coordinates: A[0.5; 0] B[0; 0] C[0.19993742518; 0.08105524265]
Centroid: CG[0.23331247506; 0.02768508088]
Coordinates of the circumscribed circle: U[0.25; -0.33217612054]
Coordinates of the inscribed circle: I[0.20219006905; 0.03992455186]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165° = 0.26217993878 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 37° = 2.49658208304 rad

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How did we calculate this triangle?

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

c = 0.5 ; ; alpha = 15° ; ; beta = 22° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 15 ° - 22 ° = 143 ° ; ;

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 = 0.5 * fraction{ sin 15° }{ sin 143° } = 0.22 ; ;

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{ 0.22**2+0.5**2 - 2 * 0.22 * 0.5 * cos 22° } ; ; b = 0.31 ; ;
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 = 0.22 ; ; b = 0.31 ; ; c = 0.5 ; ;

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

p = a+b+c = 0.22+0.31+0.5 = 1.03 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1.03 }{ 2 } = 0.51 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.51 * (0.51-0.22)(0.51-0.31)(0.51-0.5) } ; ; T = sqrt{ 0 } = 0.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.02 }{ 0.22 } = 0.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.02 }{ 0.31 } = 0.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.02 }{ 0.5 } = 0.08 ; ;

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{ 0.31**2+0.5**2-0.22**2 }{ 2 * 0.31 * 0.5 } ) = 15° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 0.22**2+0.5**2-0.31**2 }{ 2 * 0.22 * 0.5 } ) = 22° ; ; gamma = 180° - alpha - beta = 180° - 15° - 22° = 143° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.02 }{ 0.51 } = 0.04 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 0.22 }{ 2 * sin 15° } = 0.42 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.31**2+2 * 0.5**2 - 0.22**2 } }{ 2 } = 0.402 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.5**2+2 * 0.22**2 - 0.31**2 } }{ 2 } = 0.352 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.31**2+2 * 0.22**2 - 0.5**2 } }{ 2 } = 0.095 ; ;
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