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.

Right scalene triangle.

Sides: a = 0.12994095226   b = 0.03334936491   c = 0.125

Area: T = 0.00220933531
Perimeter: p = 0.28879031716
Semiperimeter: s = 0.14439515858

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 75° = 1.3098996939 rad

Height: ha = 0.03223523806
Height: hb = 0.125
Height: hc = 0.03334936491

Median: ma = 0.06547047613
Median: mb = 0.12661168352
Median: mc = 0.07109089171

Inradius: r = 0.01545420633
Circumradius: R = 0.06547047613

Vertex coordinates: A[0.125; 0] B[0; 0] C[0.125; 0.03334936491]
Centroid: CG[0.08333333333; 0.01111645497]
Coordinates of the circumscribed circle: U[0.06325; 0.01767468245]
Coordinates of the inscribed circle: I[0.11104579367; 0.01545420633]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 105° = 1.3098996939 rad

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How did we calculate this triangle?

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

c = 0.125 ; ; alpha = 90° ; ; beta = 15° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 15 ° = 75 ° ; ;

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 = 0.13 * fraction{ sin 90° }{ sin 75° } = 0.13 ; ;

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{ 0.13**2+0.13**2 - 2 * 0.13 * 0.13 * cos 15° } ; ; b = 0.03 ; ;
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 = 0.13 ; ; b = 0.03 ; ; c = 0.13 ; ;

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

p = a+b+c = 0.13+0.03+0.13 = 0.29 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 0.29 }{ 2 } = 0.14 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.14 * (0.14-0.13)(0.14-0.03)(0.14-0.13) } ; ; T = sqrt{ 0 } = 0 ; ;

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 }{ 0.13 } = 0.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0 }{ 0.03 } = 0.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0 }{ 0.13 } = 0.03 ; ;

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{ 0.03**2+0.13**2-0.13**2 }{ 2 * 0.03 * 0.13 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 0.13**2+0.13**2-0.03**2 }{ 2 * 0.13 * 0.13 } ) = 15° ; ; gamma = 180° - alpha - beta = 180° - 90° - 15° = 75° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0 }{ 0.14 } = 0.01 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 0.13 }{ 2 * sin 90° } = 0.06 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.03**2+2 * 0.13**2 - 0.13**2 } }{ 2 } = 0.065 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.13**2+2 * 0.13**2 - 0.03**2 } }{ 2 } = 0.126 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.03**2+2 * 0.13**2 - 0.13**2 } }{ 2 } = 0.071 ; ;
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