Right triangle calculator (A,a)

Please enter two properties of the right triangle

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


You have entered cathetus a and angle α.

Right scalene triangle.

Sides: a = 15000   b = 16659.18877224   c = 22417.1488248

Area: T = 124943907.918
Perimeter: p = 54076.33659704
Semiperimeter: s = 27038.16879852

Angle ∠ A = α = 42° = 0.73330382858 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 16659.18877224
Height: hb = 15000
Height: hc = 11147.17223822

Median: ma = 18269.60768806
Median: mb = 17157.56878315
Median: mc = 11208.5744124

Inradius: r = 4621.021973723
Circumradius: R = 11208.5744124

Vertex coordinates: A[22417.1488248; 0] B[0; 0] C[10036.95990954; 11147.17223822]
Centroid: CG[10818.03657811; 3715.724412739]
Coordinates of the circumscribed circle: U[11208.5744124; -0]
Coordinates of the inscribed circle: I[10378.98802628; 4621.021973723]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138° = 0.73330382858 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a angle α

a = 15000 ; ; alpha = 42° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 42 ° = 48 ° ; ;

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

 sin alpha = a:c ; ; c = a/ sin alpha = a/ sin(42 ° ) = 22417.148 ; ;

4. From side a and c we calculate b - Pythagorean theorem:

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 22417.148**2 - 15000**2 } = 16659.188 ; ;


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 = 15000 ; ; b = 16659.19 ; ; c = 22417.15 ; ;

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

p = a+b+c = 15000+16659.19+22417.15 = 54076.34 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54076.34 }{ 2 } = 27038.17 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27038.17 * (27038.17-15000)(27038.17-16659.19)(27038.17-22417.15) } ; ; T = sqrt{ 1.561 * 10**{ 16 } } = 124943907.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 124943907.92 }{ 15000 } = 16659.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 124943907.92 }{ 16659.19 } = 15000 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 124943907.92 }{ 22417.15 } = 11147.17 ; ;

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{ 15000**2-16659.19**2-22417.15**2 }{ 2 * 16659.19 * 22417.15 } ) = 42° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16659.19**2-15000**2-22417.15**2 }{ 2 * 15000 * 22417.15 } ) = 48° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22417.15**2-15000**2-16659.19**2 }{ 2 * 16659.19 * 15000 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 124943907.92 }{ 27038.17 } = 4621.02 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15000 }{ 2 * sin 42° } = 11208.57 ; ;
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