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 o.

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 and perimeter o

a = 84 ; ; o = 278 ; ;

2. From cathetus a and perimeter o we calculate cathetus b:

k_1 = p - a = b + c = 278-84 = 194 ; ; b = fraction{ k_1**2 - a**2 }{ 2 * k_1 } ; ; b = fraction{ 194**2 - 7056 }{ 2 * 194 } = 78.814 ; ;

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

c**2 = a**2+b**2 ; ; c = sqrt{ a**2+b**2 } = sqrt{ 84**2 + 78.814**2 } = sqrt{ 13267.715 } = 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 ; ;

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

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

5. Semiperimeter of the triangle

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

6. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 84 * 78.81 }{ 2 } = 3310.21 ; ;

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

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

8. 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{ 84 }{ 115.19 } ) = 46° 49'27" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 78.81 }{ 115.19 } ) = 43° 10'33" ; ; gamma = 90° ; ;

9. Inradius

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

10. Circumradius

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

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 78.81**2+2 * 115.19**2 - 84**2 } }{ 2 } = 89.307 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 115.19**2+2 * 84**2 - 78.81**2 } }{ 2 } = 92.784 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 78.81**2+2 * 84**2 - 115.19**2 } }{ 2 } = 57.593 ; ;
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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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