Right triangle calculator (A,c)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered hypotenuse c and angle α.

Right scalene triangle.

Sides: a = 42.42664068712   b = 42.42664068712   c = 60

Area: T = 900
Perimeter: p = 144.8532813742
Semiperimeter: s = 72.42664068712

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

Height: ha = 42.42664068712
Height: hb = 42.42664068712
Height: hc = 30

Median: ma = 47.43441649025
Median: mb = 47.43441649025
Median: mc = 30

Inradius: r = 12.42664068712
Circumradius: R = 30

Vertex coordinates: A[60; 0] B[0; 0] C[30; 30]
Centroid: CG[30; 10]
Coordinates of the circumscribed circle: U[30; 0]
Coordinates of the inscribed circle: I[30; 12.42664068712]

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: hypotenuse c angle α

c = 60 ; ; alpha = 45° ; ;

2. From angle α we calculate angle β:

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

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(45 ° ) = 42.426 ; ;

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{ 60**2 - 42.426**2 } = 42.426 ; ;


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 = 42.43 ; ; b = 42.43 ; ; c = 60 ; ;

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

p = a+b+c = 42.43+42.43+60 = 144.85 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 144.85 }{ 2 } = 72.43 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 72.43 * (72.43-42.43)(72.43-42.43)(72.43-60) } ; ; T = sqrt{ 810000 } = 900 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 900 }{ 42.43 } = 42.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 900 }{ 42.43 } = 42.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 900 }{ 60 } = 30 ; ;

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{ 42.43**2-42.43**2-60**2 }{ 2 * 42.43 * 60 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42.43**2-42.43**2-60**2 }{ 2 * 42.43 * 60 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 60**2-42.43**2-42.43**2 }{ 2 * 42.43 * 42.43 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 900 }{ 72.43 } = 12.43 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 42.43 }{ 2 * sin 45° } = 30 ; ;
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