Right triangle calculator (A,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, angle α and angle γ.

Right scalene triangle.

Sides: a = 1.45497474683   b = 3.5   c = 3.7888372701

Area: T = 2.53770580695
Perimeter: p = 8.73881201693
Semiperimeter: s = 4.36990600847

Angle ∠ A = α = 22.5° = 22°30' = 0.39326990817 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 3.5
Height: hb = 1.45497474683
Height: hc = 1.33993920133

Median: ma = 3.57442750217
Median: mb = 2.27325025241
Median: mc = 1.89441863505

Inradius: r = 0.58106873836
Circumradius: R = 1.89441863505

Vertex coordinates: A[3.7888372701; 0] B[0; 0] C[0.55547943372; 1.33993920133]
Centroid: CG[1.44877223461; 0.44664640044]
Coordinates of the circumscribed circle: U[1.89441863505; -0]
Coordinates of the inscribed circle: I[0.86990600847; 0.58106873836]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.5° = 157°30' = 0.39326990817 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b angle α angle γ

b = 3.5 ; ; alpha = 22.5° ; ; gamma = 90° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 22.5 ° = 67.5 ° ; ;

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

 cos alpha = b:c ; ; c = b/ cos alpha = b/ cos(22.5 ° ) = 3.788 ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(22.5 ° ) = 1.45 ; ;


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 = 1.45 ; ; b = 3.5 ; ; c = 3.79 ; ;

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

p = a+b+c = 1.45+3.5+3.79 = 8.74 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8.74 }{ 2 } = 4.37 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.37 * (4.37-1.45)(4.37-3.5)(4.37-3.79) } ; ; T = sqrt{ 6.44 } = 2.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.54 }{ 1.45 } = 3.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.54 }{ 3.5 } = 1.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.54 }{ 3.79 } = 1.34 ; ;

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{ 1.45**2-3.5**2-3.79**2 }{ 2 * 3.5 * 3.79 } ) = 22° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.5**2-1.45**2-3.79**2 }{ 2 * 1.45 * 3.79 } ) = 67° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.79**2-1.45**2-3.5**2 }{ 2 * 3.5 * 1.45 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.54 }{ 4.37 } = 0.58 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.45 }{ 2 * sin 22° 30' } = 1.89 ; ;
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