Right triangle calculator (A,a)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered cathetus a and angle α.

Right scalene triangle.

Sides: a = 52   b = 289.0043849355   c = 293.6454725718

Area: T = 7514.110008323
Perimeter: p = 634.6498575073
Semiperimeter: s = 317.3244287537

Angle ∠ A = α = 10.2° = 10°12' = 0.17880235837 rad
Angle ∠ B = β = 79.8° = 79°48' = 1.39327727431 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 289.0043849355
Height: hb = 52
Height: hc = 51.17881716144

Median: ma = 290.1711027055
Median: mb = 153.5733455504
Median: mc = 146.8222362859

Inradius: r = 23.68795618185
Circumradius: R = 146.8222362859

Vertex coordinates: A[293.6454725718; 0] B[0; 0] C[9.20884064966; 51.17881716144]
Centroid: CG[100.9511044072; 17.05993905381]
Coordinates of the circumscribed circle: U[146.8222362859; 0]
Coordinates of the inscribed circle: I[28.32204381815; 23.68795618185]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.8° = 169°48' = 0.17880235837 rad
∠ B' = β' = 100.2° = 100°12' = 1.39327727431 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a angle α

a = 52 ; ; alpha = 10.2° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 10.2 ° = 79.8 ° ; ;

3. From cathetus a and angle α we calculate hypotenuse c:

 sin alpha = a:c ; ; c = a/ sin alpha = a/ sin(10.2 ° ) = 293.645 ; ;

4. From cathetus a and hypotenuse c we calculate cathetus b - Pythagorean theorem:

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 293.645**2 - 52**2 } = 289.004 ; ;


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 = 52 ; ; b = 289 ; ; c = 293.64 ; ;

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

p = a+b+c = 52+289+293.64 = 634.65 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 634.65 }{ 2 } = 317.32 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 317.32 * (317.32-52)(317.32-289)(317.32-293.64) } ; ; T = sqrt{ 56461700.06 } = 7514.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7514.1 }{ 52 } = 289 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7514.1 }{ 289 } = 52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7514.1 }{ 293.64 } = 51.18 ; ;

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{ 52**2-289**2-293.64**2 }{ 2 * 289 * 293.64 } ) = 10° 12' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 289**2-52**2-293.64**2 }{ 2 * 52 * 293.64 } ) = 79° 48' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 293.64**2-52**2-289**2 }{ 2 * 289 * 52 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7514.1 }{ 317.32 } = 23.68 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 52 }{ 2 * sin 10° 12' } = 146.82 ; ;
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