Right triangle calculator (A)

Please enter two properties of the right triangle

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


You have entered area T and angle α.

Right scalene triangle.

Sides: a = 4.58332549729   b = 6.54655664539   c = 7.9910661196

Area: T = 15
Perimeter: p = 19.11994826227
Semiperimeter: s = 9.56597413114

Angle ∠ A = α = 35° = 0.61108652382 rad
Angle ∠ B = β = 55° = 0.96599310886 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 6.54655664539
Height: hb = 4.58332549729
Height: hc = 3.75443826805

Median: ma = 6.93551277377
Median: mb = 5.63218146451
Median: mc = 3.9955330598

Inradius: r = 1.56990801154
Circumradius: R = 3.9955330598

Vertex coordinates: A[7.9910661196; 0] B[0; 0] C[2.62988470542; 3.75443826805]
Centroid: CG[3.54398360834; 1.25114608935]
Coordinates of the circumscribed circle: U[3.9955330598; 0]
Coordinates of the inscribed circle: I[3.01441748575; 1.56990801154]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145° = 0.61108652382 rad
∠ B' = β' = 125° = 0.96599310886 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: angle α and area T

 alpha = 35° ; ; S = 15 ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 35 ° = 55 ° ; ;

3. From area T, angle α and angle β we calculate hypotenuse c:

c**2 sin alpha sin beta = 2 S ; ; c = sqrt{ fraction{ 2 S }{ sin alpha sin beta } } ; ; c = sqrt{ fraction{ 2 * 15 }{ sin 35° * sin 55° } } = 7.991 ; ;

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

S = fraction{ c * h }{ 2 } ; ; h = 2 * S / c = 2 * 15 / 7.991 = 3.754 ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = 7.991 * sin(35 ° ) = 4.583 ; ;

6. 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{ 7.991**2 - 4.583**2 } = 6.546 ; ;


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 = 4.58 ; ; b = 6.55 ; ; c = 7.99 ; ;

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

p = a+b+c = 4.58+6.55+7.99 = 19.12 ; ;

8. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.12 }{ 2 } = 9.56 ; ;

9. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 4.58 * 6.55 }{ 2 } = 15 ; ;

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

h _a = b = 6.55 ; ; h _b = a = 4.58 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15 }{ 7.99 } = 3.75 ; ;

11. 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{ 4.58 }{ 7.99 } ) = 35° ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 6.55 }{ 7.99 } ) = 55° ; ; gamma = 90° ; ;

12. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15 }{ 9.56 } = 1.57 ; ;

13. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.58 }{ 2 * sin 35° } = 4 ; ;

14. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.55**2+2 * 7.99**2 - 4.58**2 } }{ 2 } = 6.935 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.99**2+2 * 4.58**2 - 6.55**2 } }{ 2 } = 5.632 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.55**2+2 * 4.58**2 - 7.99**2 } }{ 2 } = 3.995 ; ;
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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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