Right triangle calculator (a)

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 perimeter p.

Right scalene triangle.

Sides: a = 84   b = 73.43297297297   c = 111.577027027

Area: T = 3084.049864865
Perimeter: p = 269
Semiperimeter: s = 134.5

Angle ∠ A = α = 48.84112327573° = 48°50'28″ = 0.85224403223 rad
Angle ∠ B = β = 41.15987672427° = 41°9'32″ = 0.71883560044 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 73.43297297297
Height: hb = 84
Height: hc = 55.28444165597

Median: ma = 84.59327018612
Median: mb = 91.67332311094
Median: mc = 55.78551351351

Inradius: r = 22.93297297297
Circumradius: R = 55.78551351351

Vertex coordinates: A[111.577027027; 0] B[0; 0] C[63.24326540055; 55.28444165597]
Centroid: CG[58.27109747586; 18.42881388532]
Coordinates of the circumscribed circle: U[55.78551351351; 0]
Coordinates of the inscribed circle: I[61.07702702703; 22.93297297297]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.1598767243° = 131°9'32″ = 0.85224403223 rad
∠ B' = β' = 138.8411232757° = 138°50'28″ = 0.71883560044 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a perimeter p

a = 84 ; ; p = 269 ; ;

2. From cathetus a and we calculate hypotenuse c - Pythagorean theorem:

c**2 = a**2+b**2 ; ; c = sqrt{ a**2+b**2 } = sqrt{ 84**2 + 73.43**2 } = 111.57 ; ;


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 = 84 ; ; b = 73.43 ; ; c = 111.57 ; ;

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

p = a+b+c = 84+73.43+111.57 = 269 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 269 }{ 2 } = 134.5 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 134.5 * (134.5-84)(134.5-73.43)(134.5-111.57) } ; ; T = sqrt{ 9511356.07 } = 3084.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3084.05 }{ 84 } = 73.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3084.05 }{ 73.43 } = 84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3084.05 }{ 111.57 } = 55.28 ; ;

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{ 84**2-73.43**2-111.57**2 }{ 2 * 73.43 * 111.57 } ) = 48° 50'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 73.43**2-84**2-111.57**2 }{ 2 * 84 * 111.57 } ) = 41° 9'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 111.57**2-84**2-73.43**2 }{ 2 * 73.43 * 84 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3084.05 }{ 134.5 } = 22.93 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 84 }{ 2 * sin 48° 50'28" } = 55.79 ; ;
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