Right triangle calculator (a,b)

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 cathetus b.

Right scalene triangle.

Sides: a = 40   b = 60   c = 72.11110255093

Area: T = 1200
Perimeter: p = 172.1111025509
Semiperimeter: s = 86.05655127546

Angle ∠ A = α = 33.6990067526° = 33°41'24″ = 0.58880026035 rad
Angle ∠ B = β = 56.3109932474° = 56°18'36″ = 0.98327937232 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 60
Height: hb = 40
Height: hc = 33.28220117735

Median: ma = 63.24655532034
Median: mb = 50
Median: mc = 36.05655127546

Inradius: r = 13.94444872454
Circumradius: R = 36.05655127546

Vertex coordinates: A[72.11110255093; 0] B[0; 0] C[22.1888007849; 33.28220117735]
Centroid: CG[31.43330111194; 11.09440039245]
Coordinates of the circumscribed circle: U[36.05655127546; -0]
Coordinates of the inscribed circle: I[26.05655127546; 13.94444872454]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.3109932474° = 146°18'36″ = 0.58880026035 rad
∠ B' = β' = 123.6990067526° = 123°41'24″ = 0.98327937232 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a cathetus b

a = 40 ; ; b = 60 ; ;

2. From cathetus a and cathetus b we calculate hypotenuse c - Pythagorean theorem:

c**2 = a**2+b**2 ; ; c = sqrt{ a**2+b**2 } = sqrt{ 40**2 + 60**2 } = 72.111 ; ;


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 = 40 ; ; b = 60 ; ; c = 72.11 ; ;

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

p = a+b+c = 40+60+72.11 = 172.11 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 172.11 }{ 2 } = 86.06 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 86.06 * (86.06-40)(86.06-60)(86.06-72.11) } ; ; T = sqrt{ 1440000 } = 1200 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1200 }{ 40 } = 60 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1200 }{ 60 } = 40 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1200 }{ 72.11 } = 33.28 ; ;

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{ 40**2-60**2-72.11**2 }{ 2 * 60 * 72.11 } ) = 33° 41'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-40**2-72.11**2 }{ 2 * 40 * 72.11 } ) = 56° 18'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 72.11**2-40**2-60**2 }{ 2 * 60 * 40 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1200 }{ 86.06 } = 13.94 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 40 }{ 2 * sin 33° 41'24" } = 36.06 ; ;
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