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 = 60   b = 60.2087972894   c = 85

Area: T = 1806.239918682
Perimeter: p = 205.2087972894
Semiperimeter: s = 102.6043986447

Angle ∠ A = α = 44.9010872156° = 44°54'3″ = 0.78436680561 rad
Angle ∠ B = β = 45.0999127844° = 45°5'57″ = 0.78771282707 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 60.2087972894
Height: hb = 60
Height: hc = 42.54997455722

Median: ma = 67.26881202354
Median: mb = 67.12986079105
Median: mc = 42.5

Inradius: r = 17.6043986447
Circumradius: R = 42.5

Vertex coordinates: A[85; 0] B[0; 0] C[42.35329411765; 42.54997455722]
Centroid: CG[42.45109803922; 14.16765818574]
Coordinates of the circumscribed circle: U[42.5; -0]
Coordinates of the inscribed circle: I[42.3966013553; 17.6043986447]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.0999127844° = 135°5'57″ = 0.78436680561 rad
∠ B' = β' = 134.9010872156° = 134°54'3″ = 0.78771282707 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 = 60 ; ; 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 - 60**2 } = 60.208 ; ;


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 = 60 ; ; b = 60.21 ; ; c = 85 ; ;

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

p = a+b+c = 60+60.21+85 = 205.21 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 205.21 }{ 2 } = 102.6 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.6 * (102.6-60)(102.6-60.21)(102.6-85) } ; ; T = sqrt{ 3262500 } = 1806.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1806.24 }{ 60 } = 60.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1806.24 }{ 60.21 } = 60 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1806.24 }{ 85 } = 42.5 ; ;

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{ 60**2-60.21**2-85**2 }{ 2 * 60.21 * 85 } ) = 44° 54'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60.21**2-60**2-85**2 }{ 2 * 60 * 85 } ) = 45° 5'57" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 85**2-60**2-60.21**2 }{ 2 * 60.21 * 60 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1806.24 }{ 102.6 } = 17.6 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 44° 54'3" } = 42.5 ; ;
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