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 = 90   b = 56   c = 106

Area: T = 2520
Perimeter: p = 252
Semiperimeter: s = 126

Angle ∠ A = α = 58.10992081982° = 58°6'33″ = 1.01441970088 rad
Angle ∠ B = β = 31.89107918018° = 31°53'27″ = 0.5576599318 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 56
Height: hb = 90
Height: hc = 47.54771698113

Median: ma = 71.84401002226
Median: mb = 94.25549733436
Median: mc = 53

Inradius: r = 20
Circumradius: R = 53

Vertex coordinates: A[106; 0] B[0; 0] C[76.41550943396; 47.54771698113]
Centroid: CG[60.80550314465; 15.84990566038]
Coordinates of the circumscribed circle: U[53; 0]
Coordinates of the inscribed circle: I[70; 20]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.8910791802° = 121°53'27″ = 1.01441970088 rad
∠ B' = β' = 148.1099208198° = 148°6'33″ = 0.5576599318 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 = 90 ; ; c = 106 ; ;

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{ 106**2 - 90**2 } = 56 ; ;
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 = 90 ; ; b = 56 ; ; c = 106 ; ;

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

p = a+b+c = 90+56+106 = 252 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 252 }{ 2 } = 126 ; ;

5. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 90 * 56 }{ 2 } = 2520 ; ;

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

h _a = b = 56 ; ; h _b = a = 90 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2520 }{ 106 } = 47.55 ; ;

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{ 90 }{ 106 } ) = 58° 6'33" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 56 }{ 106 } ) = 31° 53'27" ; ; gamma = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2520 }{ 126 } = 20 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 90 }{ 2 * sin 58° 6'33" } = 53 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 106**2 - 90**2 } }{ 2 } = 71.84 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 106**2+2 * 90**2 - 56**2 } }{ 2 } = 94.255 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 90**2 - 106**2 } }{ 2 } = 53 ; ;
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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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