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 = 68.41878339324   c = 91

Area: T = 2052.535501797
Perimeter: p = 219.4187833932
Semiperimeter: s = 109.7098916966

Angle ∠ A = α = 41.2549607094° = 41°14'59″ = 0.72199414589 rad
Angle ∠ B = β = 48.7550392906° = 48°45'1″ = 0.85108548679 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 68.41878339324
Height: hb = 60
Height: hc = 45.11106597356

Median: ma = 74.70660907825
Median: mb = 69.067699646
Median: mc = 45.5

Inradius: r = 18.70989169662
Circumradius: R = 45.5

Vertex coordinates: A[91; 0] B[0; 0] C[39.56604395604; 45.11106597356]
Centroid: CG[43.52201465201; 15.03768865785]
Coordinates of the circumscribed circle: U[45.5; 0]
Coordinates of the inscribed circle: I[41.29110830338; 18.70989169662]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.7550392906° = 138°45'1″ = 0.72199414589 rad
∠ B' = β' = 131.2549607094° = 131°14'59″ = 0.85108548679 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 = 91 ; ;

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{ 91**2 - 60**2 } = 68.418 ; ;


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 = 68.42 ; ; c = 91 ; ;

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

p = a+b+c = 60+68.42+91 = 219.42 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 219.42 }{ 2 } = 109.71 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 109.71 * (109.71-60)(109.71-68.42)(109.71-91) } ; ; T = sqrt{ 4212900 } = 2052.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2052.54 }{ 60 } = 68.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2052.54 }{ 68.42 } = 60 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2052.54 }{ 91 } = 45.11 ; ;

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**2-68.42**2-91**2 }{ 2 * 68.42 * 91 } ) = 41° 14'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 68.42**2-60**2-91**2 }{ 2 * 60 * 91 } ) = 48° 45'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 91**2-60**2-68.42**2 }{ 2 * 68.42 * 60 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2052.54 }{ 109.71 } = 18.71 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 41° 14'59" } = 45.5 ; ;
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