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 = 73.43297297297   c = 111.577027027

Area: T = 3084.049864865
Perimeter: p = 269
Semiperimeter: s = 134.5

Angle ∠ A = α = 48.84112327573° = 48°50'28″ = 0.85224403223 rad
Angle ∠ B = β = 41.15987672427° = 41°9'32″ = 0.71883560044 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 73.43297297297
Height: hb = 84
Height: hc = 55.28444165597

Median: ma = 84.59327018612
Median: mb = 91.67332311094
Median: mc = 55.78551351351

Inradius: r = 22.93297297297
Circumradius: R = 55.78551351351

Vertex coordinates: A[111.577027027; 0] B[0; 0] C[63.24326540055; 55.28444165597]
Centroid: CG[58.27109747586; 18.42881388532]
Coordinates of the circumscribed circle: U[55.78551351351; 0]
Coordinates of the inscribed circle: I[61.07702702703; 22.93297297297]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.1598767243° = 131°9'32″ = 0.85224403223 rad
∠ B' = β' = 138.8411232757° = 138°50'28″ = 0.71883560044 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 = 269 ; ;

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

k_1 = p - a = b + c = 269-84 = 185 ; ; b = fraction{ k_1**2 - a**2 }{ 2 * k_1 } ; ; b = fraction{ 185**2 - 7056 }{ 2 * 185 } = 73.43 ; ;

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 + 73.43**2 } = sqrt{ 12447.925 } = 111.57 ; ;
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 = 73.43 ; ; c = 111.57 ; ;

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

p = a+b+c = 84+73.43+111.57 = 269 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 269 }{ 2 } = 134.5 ; ;

6. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 84 * 73.43 }{ 2 } = 3084.05 ; ;

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

h _a = b = 73.43 ; ; h _b = a = 84 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3084.05 }{ 111.57 } = 55.28 ; ;

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 }{ 111.57 } ) = 48° 50'28" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 73.43 }{ 111.57 } ) = 41° 9'32" ; ; gamma = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3084.05 }{ 134.5 } = 22.93 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 84 }{ 2 * sin 48° 50'28" } = 55.79 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 73.43**2+2 * 111.57**2 - 84**2 } }{ 2 } = 84.593 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 111.57**2+2 * 84**2 - 73.43**2 } }{ 2 } = 91.673 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 73.43**2+2 * 84**2 - 111.57**2 } }{ 2 } = 55.785 ; ;
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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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