Right triangle calculator (a,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 a and hypotenuse c.

Right scalene triangle.

Sides: a = 56   b = 33   c = 65

Area: T = 924
Perimeter: p = 154
Semiperimeter: s = 77

Angle ∠ A = α = 59.49897625939° = 59°29'23″ = 1.03882922285 rad
Angle ∠ B = β = 30.51102374061° = 30°30'37″ = 0.53325040983 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 33
Height: hb = 56
Height: hc = 28.43107692308

Median: ma = 43.27881700168
Median: mb = 58.38802192528
Median: mc = 32.5

Inradius: r = 12
Circumradius: R = 32.5

Vertex coordinates: A[65; 0] B[0; 0] C[48.24661538462; 28.43107692308]
Centroid: CG[37.74987179487; 9.47769230769]
Coordinates of the circumscribed circle: U[32.5; -0]
Coordinates of the inscribed circle: I[44; 12]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.5110237406° = 120°30'37″ = 1.03882922285 rad
∠ B' = β' = 149.4989762594° = 149°29'23″ = 0.53325040983 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 = 56 ; ; c = 65 ; ;

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{ 65**2 - 56**2 } = 33 ; ;


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 = 56 ; ; b = 33 ; ; c = 65 ; ;

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

p = a+b+c = 56+33+65 = 154 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 154 }{ 2 } = 77 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 77 * (77-56)(77-33)(77-65) } ; ; T = sqrt{ 853776 } = 924 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 924 }{ 56 } = 33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 924 }{ 33 } = 56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 924 }{ 65 } = 28.43 ; ;

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{ 56**2-33**2-65**2 }{ 2 * 33 * 65 } ) = 59° 29'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 33**2-56**2-65**2 }{ 2 * 56 * 65 } ) = 30° 30'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 65**2-56**2-33**2 }{ 2 * 33 * 56 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 924 }{ 77 } = 12 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 56 }{ 2 * sin 59° 29'23" } = 32.5 ; ;
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