Right triangle calculator (S,b)

Please enter two properties of the right triangle

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


You have entered cathetus b and area S.

Right scalene Pythagorean triangle.

Sides: a = 36   b = 48   c = 60

Area: T = 864
Perimeter: p = 144
Semiperimeter: s = 72

Angle ∠ A = α = 36.87698976458° = 36°52'12″ = 0.64435011088 rad
Angle ∠ B = β = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 48
Height: hb = 36
Height: hc = 28.8

Median: ma = 51.26440224719
Median: mb = 43.26766153056
Median: mc = 30

Inradius: r = 12
Circumradius: R = 30

Vertex coordinates: A[60; 0] B[0; 0] C[21.6; 28.8]
Centroid: CG[27.2; 9.6]
Coordinates of the circumscribed circle: U[30; 0]
Coordinates of the inscribed circle: I[24; 12]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.1330102354° = 143°7'48″ = 0.64435011088 rad
∠ B' = β' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b and area S

b = 48 ; ; S = 864 ; ;

2. From area S and cathetus b we calculate cathetus a:

S = fraction{ ab }{ 2 } ; ; a = 2 S / b = 2 * 864/ 48 = 36 ; ;

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

c**2 = a**2+b**2 ; ; c = sqrt{ a**2+b**2 } = sqrt{ 36**2 + 48**2 } = 60 ; ;

4. From area S and hypotenuse c we calculate height h:

S = fraction{ c * h }{ 2 } ; ; h = 2 * S / c = 2 * 864 / 60 = 28.8 ; ;


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 = 36 ; ; b = 48 ; ; c = 60 ; ;

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

p = a+b+c = 36+48+60 = 144 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 144 }{ 2 } = 72 ; ;

7. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 36 * 48 }{ 2 } = 864 ; ;

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

h _a = b = 48 ; ; h _b = a = 36 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 864 }{ 60 } = 28.8 ; ;

9. Calculation of the inner angles of the triangle - basic use of sine function

 sin alpha = fraction{ a }{ c } ; ; alpha = arcsin( fraction{ a }{ c } ) = arcsin( fraction{ 36 }{ 60 } ) = 36° 52'12" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 48 }{ 60 } ) = 53° 7'48" ; ; gamma = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 864 }{ 72 } = 12 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 36 }{ 2 * sin 36° 52'12" } = 30 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 60**2 - 36**2 } }{ 2 } = 51.264 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 36**2 - 48**2 } }{ 2 } = 43.267 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 36**2 - 60**2 } }{ 2 } = 30 ; ;
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Trigonometry right triangle solver. You can calculate angles, sides (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height. A right-angled triangle is entirely determined by two independent properties. Step-by-step explanations are provided for each calculation.

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