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 = 85.87437864078   c = 120.1266213592

Area: T = 3606.699902913
Perimeter: p = 290
Semiperimeter: s = 145

Angle ∠ A = α = 44.36880266905° = 44°22'5″ = 0.77443681484 rad
Angle ∠ B = β = 45.63219733095° = 45°37'55″ = 0.79664281784 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 85.87437864078
Height: hb = 84
Height: hc = 60.04884926857

Median: ma = 95.59444935235
Median: mb = 94.33875683278
Median: mc = 60.06331067961

Inradius: r = 24.87437864078
Circumradius: R = 60.06331067961

Vertex coordinates: A[120.1266213592; 0] B[0; 0] C[58.73882203184; 60.04884926857]
Centroid: CG[59.62114779702; 20.01661642286]
Coordinates of the circumscribed circle: U[60.06331067961; 0]
Coordinates of the inscribed circle: I[59.12662135922; 24.87437864078]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.6321973309° = 135°37'55″ = 0.77443681484 rad
∠ B' = β' = 134.3688026691° = 134°22'5″ = 0.79664281784 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 = 290 ; ;

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 + 85.874**2 } = 120.126 ; ;


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 = 85.87 ; ; c = 120.13 ; ;

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

p = a+b+c = 84+85.87+120.13 = 290 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 290 }{ 2 } = 145 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 145 * (145-84)(145-85.87)(145-120.13) } ; ; T = sqrt{ 13008277.89 } = 3606.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3606.7 }{ 84 } = 85.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3606.7 }{ 85.87 } = 84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3606.7 }{ 120.13 } = 60.05 ; ;

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**2-85.87**2-120.13**2 }{ 2 * 85.87 * 120.13 } ) = 44° 22'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 85.87**2-84**2-120.13**2 }{ 2 * 84 * 120.13 } ) = 45° 37'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 120.13**2-84**2-85.87**2 }{ 2 * 85.87 * 84 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3606.7 }{ 145 } = 24.87 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 84 }{ 2 * sin 44° 22'5" } = 60.06 ; ;
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