Right triangle calculator (c,v)

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 height h.

Right scalene triangle.

Sides: a = 189.737665961   b = 63.24655532034   c = 200

Area: T = 6000
Perimeter: p = 452.9822212813
Semiperimeter: s = 226.4911106407

Angle ∠ A = α = 71.56550511771° = 71°33'54″ = 1.24990457724 rad
Angle ∠ B = β = 18.43549488229° = 18°26'6″ = 0.32217505544 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 63.24655532034
Height: hb = 189.737665961
Height: hc = 60

Median: ma = 114.018754251
Median: mb = 192.3543840617
Median: mc = 100

Inradius: r = 26.49111064067
Circumradius: R = 100

Vertex coordinates: A[200; 0] B[0; 0] C[180; 60]
Centroid: CG[126.6676666667; 20]
Coordinates of the circumscribed circle: U[100; -0]
Coordinates of the inscribed circle: I[163.2465553203; 26.49111064067]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.4354948823° = 108°26'6″ = 1.24990457724 rad
∠ B' = β' = 161.5655051177° = 161°33'54″ = 0.32217505544 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: hypotenuse c height h

c = 200 ; ; hc = 60 ; ;

2. From hypotenuse c and we calculate a,b - Pythagorean theorem, Euclid's theorem:

c = c_1+c_2 ; ; h**2 = c_1 * c_2 ; ; ; ; h**2 = c_1 * (c-c_1) ; ; h**2 = c_1 * c-c_1 **2 ; ; ; ; c_1**2 -c_1 * c + h**2 = 0 ; ; ; ; c_1**2 -200 * c_1 + 3600 = 0 ; ; ; ; c_1 = 180 ; ; c_2 = 20 ; ; ; ; a = sqrt{ c_1**2+h**2 } = sqrt{ 180**2+60**2 } = 189.737 ; ; b = sqrt{ c_2**2+h**2 } = sqrt{ 20**2+60**2 } = 63.246 ; ;


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 = 189.74 ; ; b = 63.25 ; ; c = 200 ; ;

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

p = a+b+c = 189.74+63.25+200 = 452.98 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 452.98 }{ 2 } = 226.49 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 226.49 * (226.49-189.74)(226.49-63.25)(226.49-200) } ; ; T = sqrt{ 36000000 } = 6000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6000 }{ 189.74 } = 63.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6000 }{ 63.25 } = 189.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6000 }{ 200 } = 60 ; ;

7. 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{ 189.74**2-63.25**2-200**2 }{ 2 * 63.25 * 200 } ) = 71° 33'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 63.25**2-189.74**2-200**2 }{ 2 * 189.74 * 200 } ) = 18° 26'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 200**2-189.74**2-63.25**2 }{ 2 * 63.25 * 189.74 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6000 }{ 226.49 } = 26.49 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 189.74 }{ 2 * sin 71° 33'54" } = 100 ; ;
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