Right triangle calculator (A,b)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered cathetus b and angle α.

Right scalene triangle.

Sides: a = 18.51443329744   b = 105   c = 106.6219794248

Area: T = 972.0022481155
Perimeter: p = 230.1344127222
Semiperimeter: s = 115.0677063611

Angle ∠ A = α = 10° = 0.17545329252 rad
Angle ∠ B = β = 80° = 1.39662634016 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 105
Height: hb = 18.51443329744
Height: hc = 18.2333058655

Median: ma = 105.4077282155
Median: mb = 55.66989368094
Median: mc = 53.3109897124

Inradius: r = 8.44772693632
Circumradius: R = 53.3109897124

Vertex coordinates: A[106.6219794248; 0] B[0; 0] C[3.21549801817; 18.2333058655]
Centroid: CG[36.61215914766; 6.07876862183]
Coordinates of the circumscribed circle: U[53.3109897124; -0]
Coordinates of the inscribed circle: I[10.06770636112; 8.44772693632]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170° = 0.17545329252 rad
∠ B' = β' = 100° = 1.39662634016 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b angle α

b = 105 ; ; alpha = 10° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 10 ° = 80 ° ; ;

3. From cathetus b and angle α we calculate hypotenuse c:

 cos alpha = b:c ; ; c = b/ cos alpha = b/ cos(10 ° ) = 106.62 ; ;

4. From hypotenuse c and angle α we calculate cathetus a:

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(10 ° ) = 18.514 ; ;


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 = 18.51 ; ; b = 105 ; ; c = 106.62 ; ;

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

p = a+b+c = 18.51+105+106.62 = 230.13 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 230.13 }{ 2 } = 115.07 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 115.07 * (115.07-18.51)(115.07-105)(115.07-106.62) } ; ; T = sqrt{ 944788.82 } = 972 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 972 }{ 18.51 } = 105 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 972 }{ 105 } = 18.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 972 }{ 106.62 } = 18.23 ; ;

9. 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{ 18.51**2-105**2-106.62**2 }{ 2 * 105 * 106.62 } ) = 10° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 105**2-18.51**2-106.62**2 }{ 2 * 18.51 * 106.62 } ) = 80° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 106.62**2-18.51**2-105**2 }{ 2 * 105 * 18.51 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 972 }{ 115.07 } = 8.45 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18.51 }{ 2 * sin 10° } = 53.31 ; ;
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