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 isosceles triangle.

Sides: a = 32   b = 32   c = 45.25548339959

Area: T = 512
Perimeter: p = 109.2554833996
Semiperimeter: s = 54.6277416998

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 32
Height: hb = 32
Height: hc = 22.6277416998

Median: ma = 35.777708764
Median: mb = 35.777708764
Median: mc = 22.6277416998

Inradius: r = 9.3732583002
Circumradius: R = 22.6277416998

Vertex coordinates: A[45.25548339959; 0] B[0; 0] C[22.6277416998; 22.6277416998]
Centroid: CG[22.6277416998; 7.54224723327]
Coordinates of the circumscribed circle: U[22.6277416998; -0]
Coordinates of the inscribed circle: I[22.6277416998; 9.3732583002]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 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 = 32 ; ; b = 32 ; ;

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{ 32**2 + 32**2 } = 45.255 ; ;


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 = 32 ; ; b = 32 ; ; c = 45.25 ; ;

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

p = a+b+c = 32+32+45.25 = 109.25 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 109.25 }{ 2 } = 54.63 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.63 * (54.63-32)(54.63-32)(54.63-45.25) } ; ; T = sqrt{ 262144 } = 512 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 512 }{ 32 } = 32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 512 }{ 32 } = 32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 512 }{ 45.25 } = 22.63 ; ;

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{ 32**2-32**2-45.25**2 }{ 2 * 32 * 45.25 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32**2-32**2-45.25**2 }{ 2 * 32 * 45.25 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 45.25**2-32**2-32**2 }{ 2 * 32 * 32 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 512 }{ 54.63 } = 9.37 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 32 }{ 2 * sin 45° } = 22.63 ; ;
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