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 = 8.34   b = 14.0188279666   c = 16.31215837611

Area: T = 58.45662262072
Perimeter: p = 38.67698634271
Semiperimeter: s = 19.33549317135

Angle ∠ A = α = 30.75° = 30°45' = 0.5376688745 rad
Angle ∠ B = β = 59.25° = 59°15' = 1.03441075818 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 14.0188279666
Height: hb = 8.34
Height: hc = 7.16774494719

Median: ma = 14.62553569117
Median: mb = 10.89442021827
Median: mc = 8.15657918805

Inradius: r = 3.02333479525
Circumradius: R = 8.15657918805

Vertex coordinates: A[16.31215837611; 0] B[0; 0] C[4.26441843379; 7.16774494719]
Centroid: CG[6.85985893663; 2.3899149824]
Coordinates of the circumscribed circle: U[8.15657918805; 0]
Coordinates of the inscribed circle: I[5.31766520475; 3.02333479525]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.25° = 149°15' = 0.5376688745 rad
∠ B' = β' = 120.75° = 120°45' = 1.03441075818 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 = 8.34 ; ; alpha = 30.75° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 30.75 ° = 59.25 ° ; ;

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

 sin alpha = a:c ; ; c = a/ sin alpha = 8.34/ sin(30.75 ° ) = 16.312 ; ;

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{ 16.312**2 - 8.34**2 } = 14.018 ; ;
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 = 8.34 ; ; b = 14.02 ; ; c = 16.31 ; ;

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

p = a+b+c = 8.34+14.02+16.31 = 38.67 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.67 }{ 2 } = 19.33 ; ;

7. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 8.34 * 14.02 }{ 2 } = 58.46 ; ;

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

h _a = b = 14.02 ; ; h _b = a = 8.34 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58.46 }{ 16.31 } = 7.17 ; ;

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{ 8.34 }{ 16.31 } ) = 30° 45' ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 14.02 }{ 16.31 } ) = 59° 15' ; ; gamma = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58.46 }{ 19.33 } = 3.02 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.34 }{ 2 * sin 30° 45' } = 8.16 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.02**2+2 * 16.31**2 - 8.34**2 } }{ 2 } = 14.625 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.31**2+2 * 8.34**2 - 14.02**2 } }{ 2 } = 10.894 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.02**2+2 * 8.34**2 - 16.31**2 } }{ 2 } = 8.156 ; ;
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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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