Right triangle calculator (b,c) - 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 b and hypotenuse c.

Right scalene triangle.

Sides: a = 17.32105080757   b = 10   c = 20

Area: T = 86.60325403784
Perimeter: p = 47.32105080757
Semiperimeter: s = 23.66602540378

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 10
Height: hb = 17.32105080757
Height: hc = 8.66602540378

Median: ma = 13.22987565553
Median: mb = 18.02877563773
Median: mc = 10

Inradius: r = 3.66602540378
Circumradius: R = 10

Vertex coordinates: A[20; 0] B[0; 0] C[15; 8.66602540378]
Centroid: CG[11.66766666667; 2.88767513459]
Coordinates of the circumscribed circle: U[10; 0]
Coordinates of the inscribed circle: I[13.66602540378; 3.66602540378]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b and hypotenuse c

b = 10 ; ; c = 20 ; ;

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

c**2 = a**2+b**2 ; ; a = sqrt{ c**2 - b**2 } = sqrt{ 20**2 - 10**2 } = 17.321 ; ;


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 = 17.32 ; ; b = 10 ; ; c = 20 ; ;

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

p = a+b+c = 17.32+10+20 = 47.32 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47.32 }{ 2 } = 23.66 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.66 * (23.66-17.32)(23.66-10)(23.66-20) } ; ; T = sqrt{ 7500 } = 86.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 86.6 }{ 17.32 } = 10 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 86.6 }{ 10 } = 17.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 86.6 }{ 20 } = 8.66 ; ;

7. 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{ 17.32**2-10**2-20**2 }{ 2 * 10 * 20 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-17.32**2-20**2 }{ 2 * 17.32 * 20 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-17.32**2-10**2 }{ 2 * 10 * 17.32 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 86.6 }{ 23.66 } = 3.66 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17.32 }{ 2 * sin 60° } = 10 ; ;
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