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 = 591   b = 1933.074389752   c = 2021.399943932

Area: T = 571223.3376718
Perimeter: p = 4545.473333685
Semiperimeter: s = 2272.737666842

Angle ∠ A = α = 17° = 0.29767059728 rad
Angle ∠ B = β = 73° = 1.2744090354 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1933.074389752
Height: hb = 591
Height: hc = 565.1766110774

Median: ma = 1955.52993256
Median: mb = 1132.905541235
Median: mc = 1010.769971966

Inradius: r = 251.3377229101
Circumradius: R = 1010.769971966

Vertex coordinates: A[2021.399943932; 0] B[0; 0] C[172.7921677491; 565.1766110774]
Centroid: CG[731.3977038938; 188.3922036925]
Coordinates of the circumscribed circle: U[1010.769971966; -0]
Coordinates of the inscribed circle: I[339.6632770899; 251.3377229101]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163° = 0.29767059728 rad
∠ B' = β' = 107° = 1.2744090354 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 = 591 ; ; alpha = 17° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 17 ° = 73 ° ; ;

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

 sin alpha = a:c ; ; c = a/ sin alpha = a/ sin(17 ° ) = 2021.399 ; ;

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{ 2021.399**2 - 591**2 } = 1933.074 ; ;


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 = 591 ; ; b = 1933.07 ; ; c = 2021.4 ; ;

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

p = a+b+c = 591+1933.07+2021.4 = 4545.47 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4545.47 }{ 2 } = 2272.74 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2272.74 * (2272.74-591)(2272.74-1933.07)(2272.74-2021.4) } ; ; T = sqrt{ 326296100412 } = 571223.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 571223.34 }{ 591 } = 1933.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 571223.34 }{ 1933.07 } = 591 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 571223.34 }{ 2021.4 } = 565.18 ; ;

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{ 591**2-1933.07**2-2021.4**2 }{ 2 * 1933.07 * 2021.4 } ) = 17° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1933.07**2-591**2-2021.4**2 }{ 2 * 591 * 2021.4 } ) = 73° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2021.4**2-591**2-1933.07**2 }{ 2 * 1933.07 * 591 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 571223.34 }{ 2272.74 } = 251.34 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 591 }{ 2 * sin 17° } = 1010.7 ; ;
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