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 and angle α.

Right scalene triangle.

Sides: a = 1.22004507735   b = 1.8   c = 2.16435808419

Area: T = 1.08804056961
Perimeter: p = 5.16440316154
Semiperimeter: s = 2.58220158077

Angle ∠ A = α = 33.7° = 33°42' = 0.58881759579 rad
Angle ∠ B = β = 56.3° = 56°18' = 0.98326203689 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1.8
Height: hb = 1.22004507735
Height: hc = 0.99987199694

Median: ma = 1.89774378817
Median: mb = 1.55003606432
Median: mc = 1.0821790421

Inradius: r = 0.41884349658
Circumradius: R = 1.0821790421

Vertex coordinates: A[2.16435808419; 0] B[0; 0] C[0.66660634221; 0.99987199694]
Centroid: CG[0.94332147547; 0.33329066565]
Coordinates of the circumscribed circle: U[1.0821790421; 0]
Coordinates of the inscribed circle: I[0.78220158077; 0.41884349658]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.3° = 146°18' = 0.58881759579 rad
∠ B' = β' = 123.7° = 123°42' = 0.98326203689 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

b = 1.8 ; ; alpha = 33.7° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 33.7 ° = 56.3 ° ; ;

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

 cos alpha = b:c ; ; c = b/ cos alpha = b/ cos(33.7 ° ) = 2.164 ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(33.7 ° ) = 1.2 ; ;


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 = 1.2 ; ; b = 1.8 ; ; c = 2.16 ; ;

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

p = a+b+c = 1.2+1.8+2.16 = 5.16 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5.16 }{ 2 } = 2.58 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.58 * (2.58-1.2)(2.58-1.8)(2.58-2.16) } ; ; T = sqrt{ 1.17 } = 1.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.08 }{ 1.2 } = 1.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.08 }{ 1.8 } = 1.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.08 }{ 2.16 } = 1 ; ;

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{ 1.2**2-1.8**2-2.16**2 }{ 2 * 1.8 * 2.16 } ) = 33° 42' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.8**2-1.2**2-2.16**2 }{ 2 * 1.2 * 2.16 } ) = 56° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.16**2-1.2**2-1.8**2 }{ 2 * 1.8 * 1.2 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.08 }{ 2.58 } = 0.42 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.2 }{ 2 * sin 33° 42' } = 1.08 ; ;
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