Right triangle calculator (b,c)

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 = 5.65768542495   b = 39.59879797464   c = 40

Area: T = 112
Perimeter: p = 85.25548339959
Semiperimeter: s = 42.6277416998

Angle ∠ A = α = 8.13301023542° = 8°7'48″ = 0.14218970546 rad
Angle ∠ B = β = 81.87698976458° = 81°52'12″ = 1.42988992722 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 39.59879797464
Height: hb = 5.65768542495
Height: hc = 5.6

Median: ma = 39.69988664826
Median: mb = 20.5911260282
Median: mc = 20

Inradius: r = 2.6277416998
Circumradius: R = 20

Vertex coordinates: A[40; 0] B[0; 0] C[0.8; 5.6]
Centroid: CG[13.6; 1.86766666667]
Coordinates of the circumscribed circle: U[20; 0]
Coordinates of the inscribed circle: I[3.02994372515; 2.6277416998]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 171.8769897646° = 171°52'12″ = 0.14218970546 rad
∠ B' = β' = 98.13301023542° = 98°7'48″ = 1.42988992722 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b hypotenuse c

b = 1 ; ; c = 40 ; ;

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

c**2 = a**2+b**2 ; ; a = sqrt{ c**2 - b**2 } = sqrt{ 40**2 - 39.598**2 } = 5.657 ; ;


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 = 5.66 ; ; b = 39.6 ; ; c = 40 ; ;

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

p = a+b+c = 5.66+39.6+40 = 85.25 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 85.25 }{ 2 } = 42.63 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.63 * (42.63-5.66)(42.63-39.6)(42.63-40) } ; ; T = sqrt{ 12544 } = 112 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 112 }{ 5.66 } = 39.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 112 }{ 39.6 } = 5.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 112 }{ 40 } = 5.6 ; ;

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{ 5.66**2-39.6**2-40**2 }{ 2 * 39.6 * 40 } ) = 8° 7'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 39.6**2-5.66**2-40**2 }{ 2 * 5.66 * 40 } ) = 81° 52'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 40**2-5.66**2-39.6**2 }{ 2 * 39.6 * 5.66 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 112 }{ 42.63 } = 2.63 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.66 }{ 2 * sin 8° 7'48" } = 20 ; ;
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