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 = 450   b = 499.7765631673   c = 672.5144447439

Area: T = 112449.5177126
Perimeter: p = 1622.299007911
Semiperimeter: s = 811.1455039556

Angle ∠ A = α = 42° = 0.73330382858 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 499.7765631673
Height: hb = 450
Height: hc = 334.4155171465

Median: ma = 548.0888206418
Median: mb = 514.7277034945
Median: mc = 336.257722372

Inradius: r = 138.6310592117
Circumradius: R = 336.257722372

Vertex coordinates: A[672.5144447439; 0] B[0; 0] C[301.1098772861; 334.4155171465]
Centroid: CG[324.5411073434; 111.4721723822]
Coordinates of the circumscribed circle: U[336.257722372; -0]
Coordinates of the inscribed circle: I[311.3699407883; 138.6310592117]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138° = 0.73330382858 rad
∠ B' = β' = 132° = 0.8387758041 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 β

a = 450 ; ; beta = 48° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 48 ° = 42 ° ; ;

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

 sin alpha = a:c ; ; c = a/ sin alpha = 450/ sin(42 ° ) = 672.514 ; ;

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{ 672.514**2 - 450**2 } = 499.776 ; ;
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 = 450 ; ; b = 499.78 ; ; c = 672.51 ; ;

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

p = a+b+c = 450+499.78+672.51 = 1622.29 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1622.29 }{ 2 } = 811.15 ; ;

7. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 450 * 499.78 }{ 2 } = 112449.52 ; ;

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

h _a = b = 499.78 ; ; h _b = a = 450 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 112449.52 }{ 672.51 } = 334.42 ; ;

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{ 450 }{ 672.51 } ) = 42° ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 499.78 }{ 672.51 } ) = 48° ; ; gamma = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 112449.52 }{ 811.15 } = 138.63 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 450 }{ 2 * sin 42° } = 336.26 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 499.78**2+2 * 672.51**2 - 450**2 } }{ 2 } = 548.088 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 672.51**2+2 * 450**2 - 499.78**2 } }{ 2 } = 514.727 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 499.78**2+2 * 450**2 - 672.51**2 } }{ 2 } = 336.257 ; ;
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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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