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 = 100   b = 100   c = 141.4211356237

Area: T = 5000
Perimeter: p = 341.4211356237
Semiperimeter: s = 170.7110678119

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

Height: ha = 100
Height: hb = 100
Height: hc = 70.71106781187

Median: ma = 111.8033398875
Median: mb = 111.8033398875
Median: mc = 70.71106781187

Inradius: r = 29.28993218813
Circumradius: R = 70.71106781187

Vertex coordinates: A[141.4211356237; 0] B[0; 0] C[70.71106781187; 70.71106781187]
Centroid: CG[70.71106781187; 23.57702260396]
Coordinates of the circumscribed circle: U[70.71106781187; 0]
Coordinates of the inscribed circle: I[70.71106781187; 29.28993218813]

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 = 100 ; ; b = 100 ; ;

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{ 100**2 + 100**2 } = 141.421 ; ;


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 = 100 ; ; b = 100 ; ; c = 141.42 ; ;

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

p = a+b+c = 100+100+141.42 = 341.42 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 341.42 }{ 2 } = 170.71 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 170.71 * (170.71-100)(170.71-100)(170.71-141.42) } ; ; T = sqrt{ 25000000 } = 5000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5000 }{ 100 } = 100 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5000 }{ 100 } = 100 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5000 }{ 141.42 } = 70.71 ; ;

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{ 100**2-100**2-141.42**2 }{ 2 * 100 * 141.42 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-100**2-141.42**2 }{ 2 * 100 * 141.42 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 141.42**2-100**2-100**2 }{ 2 * 100 * 100 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5000 }{ 170.71 } = 29.29 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 45° } = 70.71 ; ;
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