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 = 80   b = 84   c = 116

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

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

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

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

Inradius: r = 24
Circumradius: R = 58

Vertex coordinates: A[116; 0] B[0; 0] C[55.17224137931; 57.93110344828]
Centroid: CG[57.05774712644; 19.31103448276]
Coordinates of the circumscribed circle: U[58; 0]
Coordinates of the inscribed circle: I[56; 24]

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

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How did we calculate this triangle?

1. Input data entered: cathetus a and hypotenuse c

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

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

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

4. Semiperimeter of the triangle

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

5. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 80 * 84 }{ 2 } = 3360 ; ;

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

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

7. Calculation of the inner angles of the triangle - basic use of sine function

 sin alpha = fraction{ a }{ c } ; ; alpha = arcsin( fraction{ a }{ c } ) = arcsin( fraction{ 80 }{ 116 } ) = 43° 36'10" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 84 }{ 116 } ) = 46° 23'50" ; ; gamma = 90° ; ;

8. Inradius

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

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 80 }{ 2 * sin 43° 36'10" } = 58 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 84**2+2 * 116**2 - 80**2 } }{ 2 } = 93.038 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 116**2+2 * 80**2 - 84**2 } }{ 2 } = 90.355 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 84**2+2 * 80**2 - 116**2 } }{ 2 } = 58 ; ;
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Trigonometry right triangle solver. You can calculate angles, sides (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height. A right-angled triangle is entirely determined by two independent properties. Step-by-step explanations are provided for each calculation.

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