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 = 39   b = 18.1865998668   c = 43.03217388395

Area: T = 354.6276974027
Perimeter: p = 100.2187737508
Semiperimeter: s = 50.10988687538

Angle ∠ A = α = 65° = 1.13444640138 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 18.1865998668
Height: hb = 39
Height: hc = 16.48221122079

Median: ma = 26.66442184876
Median: mb = 40.04660065036
Median: mc = 21.51658694198

Inradius: r = 7.07771299143
Circumradius: R = 21.51658694198

Vertex coordinates: A[43.03217388395; 0] B[0; 0] C[35.34660036944; 16.48221122079]
Centroid: CG[26.1265914178; 5.49440374026]
Coordinates of the circumscribed circle: U[21.51658694198; 0]
Coordinates of the inscribed circle: I[31.92328700857; 7.07771299143]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115° = 1.13444640138 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

a = 39 ; ; beta = 25° ; ; gamma = 90° ; ;

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

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 25 ° - 90 ° = 65 ° ; ;

3. From angle β, angle α and side a we calculate 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 = 39 * fraction{ sin(25° ) }{ sin (65° ) } = 18.19 ; ;

4. From angle γ, angle α and side a we calculate 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 = 39 * fraction{ sin(90° ) }{ sin (65° ) } = 43.03 ; ;


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 = 39 ; ; b = 18.19 ; ; c = 43.03 ; ;

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

p = a+b+c = 39+18.19+43.03 = 100.22 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 100.22 }{ 2 } = 50.11 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 50.11 * (50.11-39)(50.11-18.19)(50.11-43.03) } ; ; T = sqrt{ 125760.29 } = 354.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 354.63 }{ 39 } = 18.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 354.63 }{ 18.19 } = 39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 354.63 }{ 43.03 } = 16.48 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 39**2-18.19**2-43.03**2 }{ 2 * 18.19 * 43.03 } ) = 65° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18.19**2-39**2-43.03**2 }{ 2 * 39 * 43.03 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 43.03**2-39**2-18.19**2 }{ 2 * 18.19 * 39 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 354.63 }{ 50.11 } = 7.08 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39 }{ 2 * sin 65° } = 21.52 ; ;




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