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 = 78.81444329897   c = 115.186556701

Area: T = 3310.206618557
Perimeter: p = 278
Semiperimeter: s = 139

Angle ∠ A = α = 46.82442283386° = 46°49'27″ = 0.81772369542 rad
Angle ∠ B = β = 43.17657716614° = 43°10'33″ = 0.75435593726 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 78.81444329897
Height: hb = 84
Height: hc = 57.4766058355

Median: ma = 89.30768577853
Median: mb = 92.78443128544
Median: mc = 57.59327835052

Inradius: r = 23.81444329897
Circumradius: R = 57.59327835052

Vertex coordinates: A[115.186556701; 0] B[0; 0] C[61.25876747516; 57.4766058355]
Centroid: CG[58.81444139206; 19.15986861183]
Coordinates of the circumscribed circle: U[57.59327835052; -0]
Coordinates of the inscribed circle: I[60.18655670103; 23.81444329897]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.1765771661° = 133°10'33″ = 0.81772369542 rad
∠ B' = β' = 136.8244228339° = 136°49'27″ = 0.75435593726 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 = 278 ; ;

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 + 78.814**2 } = 115.186 ; ;


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 = 78.81 ; ; c = 115.19 ; ;

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

p = a+b+c = 84+78.81+115.19 = 278 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 278 }{ 2 } = 139 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 139 * (139-84)(139-78.81)(139-115.19) } ; ; T = sqrt{ 10957464.99 } = 3310.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3310.21 }{ 84 } = 78.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3310.21 }{ 78.81 } = 84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3310.21 }{ 115.19 } = 57.48 ; ;

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**2-78.81**2-115.19**2 }{ 2 * 78.81 * 115.19 } ) = 46° 49'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 78.81**2-84**2-115.19**2 }{ 2 * 84 * 115.19 } ) = 43° 10'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 115.19**2-84**2-78.81**2 }{ 2 * 78.81 * 84 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3310.21 }{ 139 } = 23.81 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 84 }{ 2 * sin 46° 49'27" } = 57.59 ; ;
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