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 inradius r

a = 14 ; ; r = 5 ; ;

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

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

3. From and hypotenuse c we calculate h:

S = fraction{ c * h }{ 2 } ; ; h = 2 * S / c = 2 * 157.5 / c = 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 ; ;

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

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

5. Semiperimeter of the triangle

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

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-14)(31.5-22.5)(31.5-26.5) } ; ; T = sqrt{ 24806.25 } = 157.5 ; ;

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

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

8. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 14**2-22.5**2-26.5**2 }{ 2 * 22.5 * 26.5 } ) = 31° 53'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22.5**2-14**2-26.5**2 }{ 2 * 14 * 26.5 } ) = 58° 6'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 26.5**2-14**2-22.5**2 }{ 2 * 22.5 * 14 } ) = 90° ; ;

9. Inradius

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

10. Circumradius

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