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 = 110   b = 420   c = 434.1665866922

Area: T = 23100
Perimeter: p = 964.1665866922
Semiperimeter: s = 482.0832933461

Angle ∠ A = α = 14.67663931375° = 14°40'35″ = 0.25661513826 rad
Angle ∠ B = β = 75.32436068625° = 75°19'25″ = 1.31546449442 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 420
Height: hb = 110
Height: hc = 106.4110944572

Median: ma = 423.5865882673
Median: mb = 237.0655391823
Median: mc = 217.0832933461

Inradius: r = 47.91770665391
Circumradius: R = 217.0832933461

Vertex coordinates: A[434.1665866922; 0] B[0; 0] C[27.87695331021; 106.4110944572]
Centroid: CG[154.0121800008; 35.47703148573]
Coordinates of the circumscribed circle: U[217.0832933461; -0]
Coordinates of the inscribed circle: I[62.08329334609; 47.91770665391]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.3243606863° = 165°19'25″ = 0.25661513826 rad
∠ B' = β' = 104.6766393137° = 104°40'35″ = 1.31546449442 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 = 110 ; ; b = 420 ; ;

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{ 110**2 + 420**2 } = 434.166 ; ;


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 = 110 ; ; b = 420 ; ; c = 434.17 ; ;

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

p = a+b+c = 110+420+434.17 = 964.17 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 964.17 }{ 2 } = 482.08 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 482.08 * (482.08-110)(482.08-420)(482.08-434.17) } ; ; T = sqrt{ 533610000 } = 23100 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23100 }{ 110 } = 420 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23100 }{ 420 } = 110 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23100 }{ 434.17 } = 106.41 ; ;

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{ 110**2-420**2-434.17**2 }{ 2 * 420 * 434.17 } ) = 14° 40'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 420**2-110**2-434.17**2 }{ 2 * 110 * 434.17 } ) = 75° 19'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 434.17**2-110**2-420**2 }{ 2 * 420 * 110 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23100 }{ 482.08 } = 47.92 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 110 }{ 2 * sin 14° 40'35" } = 217.08 ; ;
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