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 = 100   b = 180   c = 205.913260282

Area: T = 9000
Perimeter: p = 485.913260282
Semiperimeter: s = 242.956630141

Angle ∠ A = α = 29.05546040991° = 29°3'17″ = 0.50770985044 rad
Angle ∠ B = β = 60.94553959009° = 60°56'43″ = 1.06436978224 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 180
Height: hb = 100
Height: hc = 87.41657276122

Median: ma = 186.8155416923
Median: mb = 134.5366240471
Median: mc = 102.956630141

Inradius: r = 37.04436985901
Circumradius: R = 102.956630141

Vertex coordinates: A[205.913260282; 0] B[0; 0] C[48.56442931179; 87.41657276122]
Centroid: CG[84.82656319792; 29.13985758707]
Coordinates of the circumscribed circle: U[102.956630141; -0]
Coordinates of the inscribed circle: I[62.95663014099; 37.04436985901]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.9455395901° = 150°56'43″ = 0.50770985044 rad
∠ B' = β' = 119.0554604099° = 119°3'17″ = 1.06436978224 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 = 180 ; ;

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 + 180**2 } = 205.913 ; ;


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 = 180 ; ; c = 205.91 ; ;

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

p = a+b+c = 100+180+205.91 = 485.91 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 485.91 }{ 2 } = 242.96 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 242.96 * (242.96-100)(242.96-180)(242.96-205.91) } ; ; T = sqrt{ 81000000 } = 9000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9000 }{ 100 } = 180 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9000 }{ 180 } = 100 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9000 }{ 205.91 } = 87.42 ; ;

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**2-180**2-205.91**2 }{ 2 * 180 * 205.91 } ) = 29° 3'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 180**2-100**2-205.91**2 }{ 2 * 100 * 205.91 } ) = 60° 56'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 205.91**2-100**2-180**2 }{ 2 * 180 * 100 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9000 }{ 242.96 } = 37.04 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 29° 3'17" } = 102.96 ; ;
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