Right triangle calculator (A,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 angle α.

Right scalene triangle.

Sides: a = 145   b = 261.5876924514   c = 299.0866474246

Area: T = 18965.05220273
Perimeter: p = 705.673339876
Semiperimeter: s = 352.837669938

Angle ∠ A = α = 29° = 0.50661454831 rad
Angle ∠ B = β = 61° = 1.06546508437 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 261.5876924514
Height: hb = 145
Height: hc = 126.8219857535

Median: ma = 271.4487912272
Median: mb = 195.2743986412
Median: mc = 149.5433237123

Inradius: r = 53.75502251342
Circumradius: R = 149.5433237123

Vertex coordinates: A[299.0866474246; 0] B[0; 0] C[70.29773949357; 126.8219857535]
Centroid: CG[123.1287956394; 42.27332858451]
Coordinates of the circumscribed circle: U[149.5433237123; -0]
Coordinates of the inscribed circle: I[91.25497748658; 53.75502251342]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151° = 0.50661454831 rad
∠ B' = β' = 119° = 1.06546508437 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a angle α

a = 145 ; ; alpha = 29° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 29 ° = 61 ° ; ;

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

 sin alpha = a:c ; ; c = a/ sin alpha = a/ sin(29 ° ) = 299.086 ; ;

4. 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{ 299.086**2 - 145**2 } = 261.587 ; ;


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 = 145 ; ; b = 261.59 ; ; c = 299.09 ; ;

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

p = a+b+c = 145+261.59+299.09 = 705.67 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 705.67 }{ 2 } = 352.84 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 352.84 * (352.84-145)(352.84-261.59)(352.84-299.09) } ; ; T = sqrt{ 359673198.4 } = 18965.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18965.05 }{ 145 } = 261.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18965.05 }{ 261.59 } = 145 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18965.05 }{ 299.09 } = 126.82 ; ;

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{ 145**2-261.59**2-299.09**2 }{ 2 * 261.59 * 299.09 } ) = 29° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 261.59**2-145**2-299.09**2 }{ 2 * 145 * 299.09 } ) = 61° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 299.09**2-145**2-261.59**2 }{ 2 * 261.59 * 145 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18965.05 }{ 352.84 } = 53.75 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 145 }{ 2 * sin 29° } = 149.54 ; ;
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