Right triangle calculator (B,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 angle β.

Right scalene triangle.

Sides: a = 412   b = 412   c = 582.6565987698

Area: T = 84872
Perimeter: p = 1406.65659877
Semiperimeter: s = 703.3287993849

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 412
Height: hb = 412
Height: hc = 291.3287993849

Median: ma = 460.6330003365
Median: mb = 460.6330003365
Median: mc = 291.3287993849

Inradius: r = 120.6722006151
Circumradius: R = 291.3287993849

Vertex coordinates: A[582.6565987698; 0] B[0; 0] C[291.3287993849; 291.3287993849]
Centroid: CG[291.3287993849; 97.1099331283]
Coordinates of the circumscribed circle: U[291.3287993849; -0]
Coordinates of the inscribed circle: I[291.3287993849; 120.6722006151]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b angle β

b = 412 ; ; beta = 45° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 45 ° = 45 ° ; ;

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

 cos alpha = b:c ; ; c = b/ cos alpha = b/ cos(45 ° ) = 582.656 ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(45 ° ) = 412 ; ;


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 = 412 ; ; b = 412 ; ; c = 582.66 ; ;

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

p = a+b+c = 412+412+582.66 = 1406.66 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1406.66 }{ 2 } = 703.33 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 703.33 * (703.33-412)(703.33-412)(703.33-582.66) } ; ; T = sqrt{ 7203256384 } = 84872 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 84872 }{ 412 } = 412 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 84872 }{ 412 } = 412 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 84872 }{ 582.66 } = 291.33 ; ;

9. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 412**2-412**2-582.66**2 }{ 2 * 412 * 582.66 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 412**2-412**2-582.66**2 }{ 2 * 412 * 582.66 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 582.66**2-412**2-412**2 }{ 2 * 412 * 412 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 84872 }{ 703.33 } = 120.67 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 412 }{ 2 * sin 45° } = 291.33 ; ;
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