Right triangle calculator (B,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 β.

Right scalene triangle.

Sides: a = 150   b = 40.19223788647   c = 155.2911427062

Area: T = 3014.428841485
Perimeter: p = 345.4843805926
Semiperimeter: s = 172.7421902963

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

Height: ha = 40.19223788647
Height: hb = 150
Height: hc = 38.82328567654

Median: ma = 85.09107005424
Median: mb = 151.3440202292
Median: mc = 77.64657135308

Inradius: r = 17.45504759016
Circumradius: R = 77.64657135308

Vertex coordinates: A[155.2911427062; 0] B[0; 0] C[144.8898873943; 38.82328567654]
Centroid: CG[100.0660100335; 12.94109522551]
Coordinates of the circumscribed circle: U[77.64657135308; -0]
Coordinates of the inscribed circle: I[132.5549524098; 17.45504759016]

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

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How did we calculate this triangle?

1. Input data entered: cathetus a angle β

a = 150 ; ; beta = 15° ; ;

2. From angle β we calculate angle α:

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

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

 sin alpha = a:c ; ; c = a/ sin alpha = a/ sin(75 ° ) = 155.291 ; ;

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{ 155.291**2 - 150**2 } = 40.192 ; ;


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 = 150 ; ; b = 40.19 ; ; c = 155.29 ; ;

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

p = a+b+c = 150+40.19+155.29 = 345.48 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 345.48 }{ 2 } = 172.74 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 172.74 * (172.74-150)(172.74-40.19)(172.74-155.29) } ; ; T = sqrt{ 9086778.67 } = 3014.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3014.43 }{ 150 } = 40.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3014.43 }{ 40.19 } = 150 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3014.43 }{ 155.29 } = 38.82 ; ;

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{ 150**2-40.19**2-155.29**2 }{ 2 * 40.19 * 155.29 } ) = 75° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40.19**2-150**2-155.29**2 }{ 2 * 150 * 155.29 } ) = 15° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 155.29**2-150**2-40.19**2 }{ 2 * 40.19 * 150 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3014.43 }{ 172.74 } = 17.45 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 150 }{ 2 * sin 75° } = 77.65 ; ;
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