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 = 27.5   b = 15.87771324027   c = 31.75442648054

Area: T = 218.3110570537
Perimeter: p = 75.13113972081
Semiperimeter: s = 37.56656986041

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 15.87771324027
Height: hb = 27.5
Height: hc = 13.75

Median: ma = 21.00334719352
Median: mb = 28.62329074927
Median: mc = 15.87771324027

Inradius: r = 5.81114337986
Circumradius: R = 15.87771324027

Vertex coordinates: A[31.75442648054; 0] B[0; 0] C[23.81656986041; 13.75]
Centroid: CG[18.52333211365; 4.58333333333]
Coordinates of the circumscribed circle: U[15.87771324027; 0]
Coordinates of the inscribed circle: I[21.68985662014; 5.81114337986]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 150° = 0.52435987756 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 = 27.5 ; ; beta = 30° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 30 ° = 60 ° ; ;

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

 sin alpha = a:c ; ; c = a/ sin alpha = a/ sin(60 ° ) = 31.754 ; ;

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{ 31.754**2 - 27.5**2 } = 15.877 ; ;


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 = 27.5 ; ; b = 15.88 ; ; c = 31.75 ; ;

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

p = a+b+c = 27.5+15.88+31.75 = 75.13 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 75.13 }{ 2 } = 37.57 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.57 * (37.57-27.5)(37.57-15.88)(37.57-31.75) } ; ; T = sqrt{ 47659.51 } = 218.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 218.31 }{ 27.5 } = 15.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 218.31 }{ 15.88 } = 27.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 218.31 }{ 31.75 } = 13.75 ; ;

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{ 27.5**2-15.88**2-31.75**2 }{ 2 * 15.88 * 31.75 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15.88**2-27.5**2-31.75**2 }{ 2 * 27.5 * 31.75 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 31.75**2-27.5**2-15.88**2 }{ 2 * 15.88 * 27.5 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 218.31 }{ 37.57 } = 5.81 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27.5 }{ 2 * sin 60° } = 15.88 ; ;
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