Triangle calculator

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

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

Right scalene triangle.

Sides: a = 36   b = 12.31327251597   c = 33.82989343483

Area: T = 208.2633185538
Perimeter: p = 82.1421659508
Semiperimeter: s = 41.0710829754

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 70° = 1.22217304764 rad

Height: ha = 11.57701769744
Height: hb = 33.82989343483
Height: hc = 12.31327251597

Median: ma = 18
Median: mb = 34.38545546628
Median: mc = 20.92113384047

Inradius: r = 5.0710829754
Circumradius: R = 18

Vertex coordinates: A[33.82989343483; 0] B[0; 0] C[33.82989343483; 12.31327251597]
Centroid: CG[22.55326228989; 4.10442417199]
Coordinates of the circumscribed circle: U[16.91444671741; 6.15663625799]
Coordinates of the inscribed circle: I[28.75881045943; 5.0710829754]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 110° = 1.22217304764 rad

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How did we calculate this triangle?

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

a = 36 ; ; alpha = 90° ; ; beta = 20° ; ; gamma = 70° ; ;

2. 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 = 36 * fraction{ sin 20° }{ sin 90° } = 12.31 ; ;

3. 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 = 36 * fraction{ sin 70° }{ sin 90° } = 33.83 ; ;
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 ; ; b = 12.31 ; ; c = 33.83 ; ;

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

p = a+b+c = 36+12.31+33.83 = 82.14 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 82.14 }{ 2 } = 41.07 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.07 * (41.07-36)(41.07-12.31)(41.07-33.83) } ; ; T = sqrt{ 43373.55 } = 208.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 208.26 }{ 36 } = 11.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 208.26 }{ 12.31 } = 33.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 208.26 }{ 33.83 } = 12.31 ; ;

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{ 12.31**2+33.83**2-36**2 }{ 2 * 12.31 * 33.83 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 36**2+33.83**2-12.31**2 }{ 2 * 36 * 33.83 } ) = 20° ; ; gamma = 180° - alpha - beta = 180° - 90° - 20° = 70° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 208.26 }{ 41.07 } = 5.07 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 36 }{ 2 * sin 90° } = 18 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.31**2+2 * 33.83**2 - 36**2 } }{ 2 } = 18 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 33.83**2+2 * 36**2 - 12.31**2 } }{ 2 } = 34.385 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.31**2+2 * 36**2 - 33.83**2 } }{ 2 } = 20.921 ; ;
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