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 = 5.5   b = 5.5   c = 7.77881745931

Area: T = 15.125
Perimeter: p = 18.77881745931
Semiperimeter: s = 9.38990872965

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

Height: ha = 5.5
Height: hb = 5.5
Height: hc = 3.88990872965

Median: ma = 6.14991869381
Median: mb = 6.14991869381
Median: mc = 3.88990872965

Inradius: r = 1.61109127035
Circumradius: R = 3.88990872965

Vertex coordinates: A[7.77881745931; 0] B[0; 0] C[3.88990872965; 3.88990872965]
Centroid: CG[3.88990872965; 1.29663624322]
Coordinates of the circumscribed circle: U[3.88990872965; 0]
Coordinates of the inscribed circle: I[3.88990872965; 1.61109127035]

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 = 5.5 ; ; b = 5.5 ; ;

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{ 5.5**2 + 5.5**2 } = 7.778 ; ;


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 = 5.5 ; ; b = 5.5 ; ; c = 7.78 ; ;

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

p = a+b+c = 5.5+5.5+7.78 = 18.78 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.78 }{ 2 } = 9.39 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.39 * (9.39-5.5)(9.39-5.5)(9.39-7.78) } ; ; T = sqrt{ 228.77 } = 15.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15.13 }{ 5.5 } = 5.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15.13 }{ 5.5 } = 5.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15.13 }{ 7.78 } = 3.89 ; ;

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{ 5.5**2-5.5**2-7.78**2 }{ 2 * 5.5 * 7.78 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.5**2-5.5**2-7.78**2 }{ 2 * 5.5 * 7.78 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.78**2-5.5**2-5.5**2 }{ 2 * 5.5 * 5.5 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15.13 }{ 9.39 } = 1.61 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.5 }{ 2 * sin 45° } = 3.89 ; ;
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