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 = 36   b = 20.78546096908   c = 41.56992193817

Area: T = 374.1232974435
Perimeter: p = 98.35438290725
Semiperimeter: s = 49.17769145362

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

Height: ha = 20.78546096908
Height: hb = 36
Height: hc = 18

Median: ma = 27.49554541697
Median: mb = 37.47699879904
Median: mc = 20.78546096908

Inradius: r = 7.60876951546
Circumradius: R = 20.78546096908

Vertex coordinates: A[41.56992193817; 0] B[0; 0] C[31.17769145362; 18]
Centroid: CG[24.2498711306; 6]
Coordinates of the circumscribed circle: U[20.78546096908; 0]
Coordinates of the inscribed circle: I[28.39223048454; 7.60876951546]

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 and angle α

a = 36 ; ; alpha = 60° ; ;

2. From angle α we calculate angle β:

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

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

 sin alpha = a:c ; ; c = a/ sin alpha = 36/ sin(60 ° ) = 41.569 ; ;

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{ 41.569**2 - 36**2 } = 20.785 ; ;


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 = 20.78 ; ; c = 41.57 ; ;

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

p = a+b+c = 36+20.78+41.57 = 98.35 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 98.35 }{ 2 } = 49.18 ; ;

7. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 36 * 20.78 }{ 2 } = 374.12 ; ;

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

h _a = b = 20.78 ; ; h _b = a = 36 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 374.12 }{ 41.57 } = 18 ; ;

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 }{ 41.57 } ) = 60° ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 20.78 }{ 41.57 } ) = 30° ; ; gamma = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 374.12 }{ 49.18 } = 7.61 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 36 }{ 2 * sin 60° } = 20.78 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.78**2+2 * 41.57**2 - 36**2 } }{ 2 } = 27.495 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 41.57**2+2 * 36**2 - 20.78**2 } }{ 2 } = 37.47 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.78**2+2 * 36**2 - 41.57**2 } }{ 2 } = 20.785 ; ;
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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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