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, angle α and angle γ.

Right scalene triangle.

Sides: a = 95.45994154602   b = 95.45994154602   c = 135

Area: T = 4556.25
Perimeter: p = 325.919883092
Semiperimeter: s = 162.959941546

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

Height: ha = 95.45994154602
Height: hb = 95.45994154602
Height: hc = 67.5

Median: ma = 106.7276871031
Median: mb = 106.7276871031
Median: mc = 67.5

Inradius: r = 27.95994154602
Circumradius: R = 67.5

Vertex coordinates: A[135; 0] B[0; 0] C[67.5; 67.5]
Centroid: CG[67.5; 22.5]
Coordinates of the circumscribed circle: U[67.5; -0]
Coordinates of the inscribed circle: I[67.5; 27.95994154602]

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 and angle γ

c = 135 ; ; alpha = 45° ; ; gamma = 90° ; ;

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 = 135 * sin(45 ° ) = 95.459 ; ;

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{ 135**2 - 95.459**2 } = 95.459 ; ;


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 = 95.46 ; ; b = 95.46 ; ; c = 135 ; ;

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

p = a+b+c = 95.46+95.46+135 = 325.92 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 325.92 }{ 2 } = 162.96 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 162.96 * (162.96-95.46)(162.96-95.46)(162.96-135) } ; ; T = sqrt{ 20759414.06 } = 4556.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4556.25 }{ 95.46 } = 95.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4556.25 }{ 95.46 } = 95.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4556.25 }{ 135 } = 67.5 ; ;

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{ 95.46**2-95.46**2-135**2 }{ 2 * 95.46 * 135 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 95.46**2-95.46**2-135**2 }{ 2 * 95.46 * 135 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 135**2-95.46**2-95.46**2 }{ 2 * 95.46 * 95.46 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4556.25 }{ 162.96 } = 27.96 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 95.46 }{ 2 * sin 45° } = 67.5 ; ;
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