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 cathetus a and hypotenuse c.

Right scalene triangle.

Sides: a = 77   b = 36   c = 85

Area: T = 1386
Perimeter: p = 198
Semiperimeter: s = 99

Angle ∠ A = α = 64.94223845817° = 64°56'33″ = 1.1333458435 rad
Angle ∠ B = β = 25.05876154183° = 25°3'27″ = 0.43773378917 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 36
Height: hb = 77
Height: hc = 32.61217647059

Median: ma = 52.70991073724
Median: mb = 79.07659128939
Median: mc = 42.5

Inradius: r = 14
Circumradius: R = 42.5

Vertex coordinates: A[85; 0] B[0; 0] C[69.75329411765; 32.61217647059]
Centroid: CG[51.58443137255; 10.87105882353]
Coordinates of the circumscribed circle: U[42.5; 0]
Coordinates of the inscribed circle: I[63; 14]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.0587615418° = 115°3'27″ = 1.1333458435 rad
∠ B' = β' = 154.9422384582° = 154°56'33″ = 0.43773378917 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a hypotenuse c

a = 77 ; ; c = 85 ; ;

2. 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{ 85**2 - 77**2 } = 36 ; ;


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 = 77 ; ; b = 36 ; ; c = 85 ; ;

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

p = a+b+c = 77+36+85 = 198 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 198 }{ 2 } = 99 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 99 * (99-77)(99-36)(99-85) } ; ; T = sqrt{ 1920996 } = 1386 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1386 }{ 77 } = 36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1386 }{ 36 } = 77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1386 }{ 85 } = 32.61 ; ;

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{ 77**2-36**2-85**2 }{ 2 * 36 * 85 } ) = 64° 56'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 36**2-77**2-85**2 }{ 2 * 77 * 85 } ) = 25° 3'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 85**2-77**2-36**2 }{ 2 * 36 * 77 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1386 }{ 99 } = 14 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 77 }{ 2 * sin 64° 56'33" } = 42.5 ; ;
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