Triangle calculator

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

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

Right scalene triangle.

Sides: a = 1.375   b = 1.73331495693   c = 1.05550746085

Area: T = 0.72553637933
Perimeter: p = 4.16332241777
Semiperimeter: s = 2.08216120889

Angle ∠ A = α = 52.5° = 52°30' = 0.91662978573 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 37.5° = 37°30' = 0.65444984695 rad

Height: ha = 1.05550746085
Height: hb = 0.83770469649
Height: hc = 1.375

Median: ma = 1.25993008693
Median: mb = 0.86765747846
Median: mc = 1.4732725571

Inradius: r = 0.34884625196
Circumradius: R = 0.86765747846

Vertex coordinates: A[1.05550746085; 0] B[0; 0] C[-0; 1.375]
Centroid: CG[0.35216915362; 0.45883333333]
Coordinates of the circumscribed circle: U[0.52875373042; 0.68875]
Coordinates of the inscribed circle: I[0.34884625196; 0.34884625196]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.5° = 127°30' = 0.91662978573 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 142.5° = 142°30' = 0.65444984695 rad

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How did we calculate this triangle?

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

a = 1.375 ; ; beta = 90° ; ; gamma = 37.5° ; ;

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

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 90 ° - 37.5 ° = 52.5 ° ; ;

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 = 1.38 * fraction{ sin 90° }{ sin 52° 30' } = 1.73 ; ;

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 = 1.38 * fraction{ sin 37° 30' }{ sin 52° 30' } = 1.06 ; ;
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 = 1.38 ; ; b = 1.73 ; ; c = 1.06 ; ;

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

p = a+b+c = 1.38+1.73+1.06 = 4.16 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4.16 }{ 2 } = 2.08 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.08 * (2.08-1.38)(2.08-1.73)(2.08-1.06) } ; ; T = sqrt{ 0.53 } = 0.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.73 }{ 1.38 } = 1.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.73 }{ 1.73 } = 0.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.73 }{ 1.06 } = 1.38 ; ;

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{ 1.73**2+1.06**2-1.38**2 }{ 2 * 1.73 * 1.06 } ) = 52° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.38**2+1.06**2-1.73**2 }{ 2 * 1.38 * 1.06 } ) = 90° ; ;
 gamma = 180° - alpha - beta = 180° - 52° 30' - 90° = 37° 30' ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.73 }{ 2.08 } = 0.35 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.38 }{ 2 * sin 52° 30' } = 0.87 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.73**2+2 * 1.06**2 - 1.38**2 } }{ 2 } = 1.259 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.06**2+2 * 1.38**2 - 1.73**2 } }{ 2 } = 0.867 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.73**2+2 * 1.38**2 - 1.06**2 } }{ 2 } = 1.473 ; ;
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