Right triangle calculator (a,r) - result

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 inradius r.

Right scalene triangle.

Sides: a = 14   b = 22.5   c = 26.5

Area: T = 157.5
Perimeter: p = 63
Semiperimeter: s = 31.5

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

Height: ha = 22.5
Height: hb = 14
Height: hc = 11.88767924528

Median: ma = 23.56437433359
Median: mb = 17.96600250557
Median: mc = 13.25

Inradius: r = 5
Circumradius: R = 13.25

Vertex coordinates: A[26.5; 0] B[0; 0] C[7.39662264151; 11.88767924528]
Centroid: CG[11.29987421384; 3.96222641509]
Coordinates of the circumscribed circle: U[13.25; 0]
Coordinates of the inscribed circle: I[9; 5]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.1099208198° = 148°6'33″ = 0.5576599318 rad
∠ B' = β' = 121.8910791802° = 121°53'27″ = 1.01441970088 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a and inradius r

a = 14 ; ; r = 5 ; ;

2. From cathetus a and inradius r we calculate cathetus b:

b = fraction{ 2r(a-r) }{ a-2r } ; ; b = fraction{ 2 * 5 * (14-5) }{ 14-2 * 5 } = 22.5 ; ;

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

c**2 = a**2+b**2 ; ; c = sqrt{ a**2+b**2 } = sqrt{ 14**2 + 22.5**2 } = sqrt{ 702.25 } = 26.5 ; ;

4. From hypotenuse c and inradius r we calculate area S:

T = r(c+r) = 5 (26.5+5) = 157.5 ; ;

5. From area S and hypotenuse c we calculate height h:

T = fraction{ c * h }{ 2 } ; ; h = 2 * T / c = 2 * 157.5 / 26.5 = 11.887 ; ;
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 = 14 ; ; b = 22.5 ; ; c = 26.5 ; ;

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

p = a+b+c = 14+22.5+26.5 = 63 ; ;

7. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

8. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 14 * 22.5 }{ 2 } = 157.5 ; ;

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

h _a = b = 22.5 ; ; h _b = a = 14 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 157.5 }{ 26.5 } = 11.89 ; ;

10. 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{ 14 }{ 26.5 } ) = 31° 53'27" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 22.5 }{ 26.5 } ) = 58° 6'33" ; ; gamma = 90° ; ;

11. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 157.5 }{ 31.5 } = 5 ; ;

12. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 14 }{ 2 * sin 31° 53'27" } = 13.25 ; ;

13. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.5**2+2 * 26.5**2 - 14**2 } }{ 2 } = 23.564 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.5**2+2 * 14**2 - 22.5**2 } }{ 2 } = 17.96 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.5**2+2 * 14**2 - 26.5**2 } }{ 2 } = 13.25 ; ;
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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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