Right triangle calculator (a,c)

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 hypotenuse c.

Right scalene triangle.

Sides: a = 84   b = 80   c = 116

Area: T = 3360
Perimeter: p = 280
Semiperimeter: s = 140

Angle ∠ A = α = 46.39771810273° = 46°23'50″ = 0.81097835726 rad
Angle ∠ B = β = 43.60328189727° = 43°36'10″ = 0.76110127542 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 80
Height: hb = 84
Height: hc = 57.93110344828

Median: ma = 90.35548559846
Median: mb = 93.03876267969
Median: mc = 58

Inradius: r = 24
Circumradius: R = 58

Vertex coordinates: A[116; 0] B[0; 0] C[60.82875862069; 57.93110344828]
Centroid: CG[58.94325287356; 19.31103448276]
Coordinates of the circumscribed circle: U[58; 0]
Coordinates of the inscribed circle: I[60; 24]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.6032818973° = 133°36'10″ = 0.81097835726 rad
∠ B' = β' = 136.3977181027° = 136°23'50″ = 0.76110127542 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a hypotenuse c

a = 84 ; ; c = 116 ; ;

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

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 116**2 - 84**2 } = 80 ; ;


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 = 80 ; ; c = 116 ; ;

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

p = a+b+c = 84+80+116 = 280 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 280 }{ 2 } = 140 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 140 * (140-84)(140-80)(140-116) } ; ; T = sqrt{ 11289600 } = 3360 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3360 }{ 84 } = 80 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3360 }{ 80 } = 84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3360 }{ 116 } = 57.93 ; ;

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**2-80**2-116**2 }{ 2 * 80 * 116 } ) = 46° 23'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 80**2-84**2-116**2 }{ 2 * 84 * 116 } ) = 43° 36'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 116**2-84**2-80**2 }{ 2 * 80 * 84 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3360 }{ 140 } = 24 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 84 }{ 2 * sin 46° 23'50" } = 58 ; ;
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