Triangle calculator

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

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

Acute scalene triangle.

Sides: a = 13.6   b = 13.417703685   c = 6.76768446247

Area: T = 44.23220180725
Perimeter: p = 33.78438814747
Semiperimeter: s = 16.89219407373

Angle ∠ A = α = 77° = 1.3443903524 rad
Angle ∠ B = β = 74° = 1.29215436465 rad
Angle ∠ C = γ = 29° = 0.50661454831 rad

Height: ha = 6.50547085401
Height: hb = 6.59334108354
Height: hc = 13.07331590648

Median: ma = 8.16547738489
Median: mb = 8.38987349242
Median: mc = 13.07882602961

Inradius: r = 2.6198527898
Circumradius: R = 6.9798867933

Vertex coordinates: A[6.76768446247; 0] B[0; 0] C[3.74986680391; 13.07331590648]
Centroid: CG[3.50551708879; 4.35877196883]
Coordinates of the circumscribed circle: U[3.38334223123; 6.10438554277]
Coordinates of the inscribed circle: I[3.47549038873; 2.6198527898]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103° = 1.3443903524 rad
∠ B' = β' = 106° = 1.29215436465 rad
∠ C' = γ' = 151° = 0.50661454831 rad

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How did we calculate this triangle?

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

a = 13.6 ; ; alpha = 77° ; ; beta = 74° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 77 ° - 74 ° = 29 ° ; ;

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 = 13.6 * fraction{ sin 74° }{ sin 77° } = 13.42 ; ;

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 = 13.6 * fraction{ sin 29° }{ sin 77° } = 6.77 ; ;
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 = 13.6 ; ; b = 13.42 ; ; c = 6.77 ; ;

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

p = a+b+c = 13.6+13.42+6.77 = 33.78 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.78 }{ 2 } = 16.89 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.89 * (16.89-13.6)(16.89-13.42)(16.89-6.77) } ; ; T = sqrt{ 1956.47 } = 44.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44.23 }{ 13.6 } = 6.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44.23 }{ 13.42 } = 6.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44.23 }{ 6.77 } = 13.07 ; ;

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{ 13.42**2+6.77**2-13.6**2 }{ 2 * 13.42 * 6.77 } ) = 77° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.6**2+6.77**2-13.42**2 }{ 2 * 13.6 * 6.77 } ) = 74° ; ; gamma = 180° - alpha - beta = 180° - 77° - 74° = 29° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44.23 }{ 16.89 } = 2.62 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.6 }{ 2 * sin 77° } = 6.98 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.42**2+2 * 6.77**2 - 13.6**2 } }{ 2 } = 8.165 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.77**2+2 * 13.6**2 - 13.42**2 } }{ 2 } = 8.389 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.42**2+2 * 13.6**2 - 6.77**2 } }{ 2 } = 13.078 ; ;
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