Right triangle calculator

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 hypotenuse c.

Right scalene triangle.

Sides: a = 5.12   b = 3.49765125482   c = 6.2

Area: T = 8.95110721235
Perimeter: p = 14.81765125482
Semiperimeter: s = 7.40882562741

Angle ∠ A = α = 55.67703385075° = 55°40'13″ = 0.97216307027 rad
Angle ∠ B = β = 34.33296614925° = 34°19'47″ = 0.59991656241 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 3.49765125482
Height: hb = 5.12
Height: hc = 2.88774426205

Median: ma = 4.33334974328
Median: mb = 5.41102495321
Median: mc = 3.1

Inradius: r = 1.20882562741
Circumradius: R = 3.1

Vertex coordinates: A[6.2; 0] B[0; 0] C[4.22881290323; 2.88774426205]
Centroid: CG[3.47660430108; 0.96224808735]
Coordinates of the circumscribed circle: U[3.1; -0]
Coordinates of the inscribed circle: I[3.91217437259; 1.20882562741]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.3329661492° = 124°19'47″ = 0.97216307027 rad
∠ B' = β' = 145.6770338508° = 145°40'13″ = 0.59991656241 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a hypotenuse c

a = 5.12 ; ; c = 6.2 ; ;

2. 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{ 6.2**2 - 5.12**2 } = 3.497 ; ;


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 = 5.12 ; ; b = 3.5 ; ; c = 6.2 ; ;

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

p = a+b+c = 5.12+3.5+6.2 = 14.82 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.82 }{ 2 } = 7.41 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.41 * (7.41-5.12)(7.41-3.5)(7.41-6.2) } ; ; T = sqrt{ 80.12 } = 8.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.95 }{ 5.12 } = 3.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.95 }{ 3.5 } = 5.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.95 }{ 6.2 } = 2.89 ; ;

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{ 5.12**2-3.5**2-6.2**2 }{ 2 * 3.5 * 6.2 } ) = 55° 40'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.5**2-5.12**2-6.2**2 }{ 2 * 5.12 * 6.2 } ) = 34° 19'47" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.2**2-5.12**2-3.5**2 }{ 2 * 3.5 * 5.12 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.95 }{ 7.41 } = 1.21 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.12 }{ 2 * sin 55° 40'13" } = 3.1 ; ;
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