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 = 4.38439526802   b = 31.19334441654   c = 31.5

Area: T = 68.37552915774
Perimeter: p = 67.07773968456
Semiperimeter: s = 33.53986984228

Angle ∠ A = α = 8° = 0.14396263402 rad
Angle ∠ B = β = 82° = 1.43111699866 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 31.19334441654
Height: hb = 4.38439526802
Height: hc = 4.34112883541

Median: ma = 31.27703648711
Median: mb = 16.20111351709
Median: mc = 15.75

Inradius: r = 2.03986984228
Circumradius: R = 15.75

Vertex coordinates: A[31.5; 0] B[0; 0] C[0.6110128289; 4.34112883541]
Centroid: CG[10.70333760963; 1.4477096118]
Coordinates of the circumscribed circle: U[15.75; 0]
Coordinates of the inscribed circle: I[2.34552542574; 2.03986984228]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 172° = 0.14396263402 rad
∠ B' = β' = 98° = 1.43111699866 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 = 31.5 ; ; alpha = 8° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 8 ° = 82 ° ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(8 ° ) = 4.384 ; ;

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{ 31.5**2 - 4.384**2 } = 31.193 ; ;


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 = 4.38 ; ; b = 31.19 ; ; c = 31.5 ; ;

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

p = a+b+c = 4.38+31.19+31.5 = 67.08 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67.08 }{ 2 } = 33.54 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.54 * (33.54-4.38)(33.54-31.19)(33.54-31.5) } ; ; T = sqrt{ 4675.18 } = 68.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 68.38 }{ 4.38 } = 31.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 68.38 }{ 31.19 } = 4.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 68.38 }{ 31.5 } = 4.34 ; ;

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{ 4.38**2-31.19**2-31.5**2 }{ 2 * 31.19 * 31.5 } ) = 8° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31.19**2-4.38**2-31.5**2 }{ 2 * 4.38 * 31.5 } ) = 82° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 31.5**2-4.38**2-31.19**2 }{ 2 * 31.19 * 4.38 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 68.38 }{ 33.54 } = 2.04 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.38 }{ 2 * sin 8° } = 15.75 ; ;
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