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 = 3.5   b = 3.5   c = 4.95497474683

Area: T = 6.125
Perimeter: p = 11.95497474683
Semiperimeter: s = 5.97548737342

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

Height: ha = 3.5
Height: hb = 3.5
Height: hc = 2.47548737342

Median: ma = 3.91331189606
Median: mb = 3.91331189606
Median: mc = 2.47548737342

Inradius: r = 1.02551262658
Circumradius: R = 2.47548737342

Vertex coordinates: A[4.95497474683; 0] B[0; 0] C[2.47548737342; 2.47548737342]
Centroid: CG[2.47548737342; 0.82549579114]
Coordinates of the circumscribed circle: U[2.47548737342; 0]
Coordinates of the inscribed circle: I[2.47548737342; 1.02551262658]

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 = 3.5 ; ; b = 3.5 ; ;

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{ 3.5**2 + 3.5**2 } = 4.95 ; ;


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 = 3.5 ; ; b = 3.5 ; ; c = 4.95 ; ;

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

p = a+b+c = 3.5+3.5+4.95 = 11.95 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.95 }{ 2 } = 5.97 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.97 * (5.97-3.5)(5.97-3.5)(5.97-4.95) } ; ; T = sqrt{ 37.52 } = 6.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.13 }{ 3.5 } = 3.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.13 }{ 3.5 } = 3.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.13 }{ 4.95 } = 2.47 ; ;

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{ 3.5**2-3.5**2-4.95**2 }{ 2 * 3.5 * 4.95 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.5**2-3.5**2-4.95**2 }{ 2 * 3.5 * 4.95 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.95**2-3.5**2-3.5**2 }{ 2 * 3.5 * 3.5 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.13 }{ 5.97 } = 1.03 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.5 }{ 2 * sin 45° } = 2.47 ; ;
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