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 = 3.77662317747   b = 13.16992852344   c = 13.7

Area: T = 24.8655136676
Perimeter: p = 30.6465517009
Semiperimeter: s = 15.32327585045

Angle ∠ A = α = 16° = 0.27992526803 rad
Angle ∠ B = β = 74° = 1.29215436465 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 13.16992852344
Height: hb = 3.77662317747
Height: hc = 3.632994696

Median: ma = 13.30439488569
Median: mb = 7.59106155753
Median: mc = 6.85

Inradius: r = 1.62327585045
Circumradius: R = 6.85

Vertex coordinates: A[13.7; 0] B[0; 0] C[1.04108705413; 3.632994696]
Centroid: CG[4.91436235138; 1.210998232]
Coordinates of the circumscribed circle: U[6.85; 0]
Coordinates of the inscribed circle: I[2.15334732702; 1.62327585045]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164° = 0.27992526803 rad
∠ B' = β' = 106° = 1.29215436465 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 = 13.7 ; ; alpha = 16° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 16 ° = 74 ° ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(16 ° ) = 3.776 ; ;

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{ 13.7**2 - 3.776**2 } = 13.169 ; ;


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 = 3.78 ; ; b = 13.17 ; ; c = 13.7 ; ;

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

p = a+b+c = 3.78+13.17+13.7 = 30.65 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.65 }{ 2 } = 15.32 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.32 * (15.32-3.78)(15.32-13.17)(15.32-13.7) } ; ; T = sqrt{ 618.28 } = 24.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24.87 }{ 3.78 } = 13.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24.87 }{ 13.17 } = 3.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24.87 }{ 13.7 } = 3.63 ; ;

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{ 3.78**2-13.17**2-13.7**2 }{ 2 * 13.17 * 13.7 } ) = 16° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13.17**2-3.78**2-13.7**2 }{ 2 * 3.78 * 13.7 } ) = 74° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13.7**2-3.78**2-13.17**2 }{ 2 * 13.17 * 3.78 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24.87 }{ 15.32 } = 1.62 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.78 }{ 2 * sin 16° } = 6.85 ; ;
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