Right triangle calculator (a,b)

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 cathetus b.

Right scalene triangle.

Sides: a = 61   b = 39   c = 72.40216574396

Area: T = 1189.5
Perimeter: p = 172.402165744
Semiperimeter: s = 86.20108287198

Angle ∠ A = α = 57.40774185274° = 57°24'27″ = 1.00219484684 rad
Angle ∠ B = β = 32.59325814726° = 32°35'33″ = 0.56988478584 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 39
Height: hb = 61
Height: hc = 32.85883638018

Median: ma = 49.51100999797
Median: mb = 64.04110024906
Median: mc = 36.20108287198

Inradius: r = 13.79991712802
Circumradius: R = 36.20108287198

Vertex coordinates: A[72.40216574396; 0] B[0; 0] C[51.39438510745; 32.85883638018]
Centroid: CG[41.26551695047; 10.95327879339]
Coordinates of the circumscribed circle: U[36.20108287198; -0]
Coordinates of the inscribed circle: I[47.20108287198; 13.79991712802]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122.5932581473° = 122°35'33″ = 1.00219484684 rad
∠ B' = β' = 147.4077418527° = 147°24'27″ = 0.56988478584 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a cathetus b

a = 61 ; ; b = 39 ; ;

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

c**2 = a**2+b**2 ; ; c = sqrt{ a**2+b**2 } = sqrt{ 61**2 + 39**2 } = 72.402 ; ;


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 = 61 ; ; b = 39 ; ; c = 72.4 ; ;

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

p = a+b+c = 61+39+72.4 = 172.4 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 172.4 }{ 2 } = 86.2 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 86.2 * (86.2-61)(86.2-39)(86.2-72.4) } ; ; T = sqrt{ 1414910.25 } = 1189.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1189.5 }{ 61 } = 39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1189.5 }{ 39 } = 61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1189.5 }{ 72.4 } = 32.86 ; ;

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{ 61**2-39**2-72.4**2 }{ 2 * 39 * 72.4 } ) = 57° 24'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 39**2-61**2-72.4**2 }{ 2 * 61 * 72.4 } ) = 32° 35'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 72.4**2-61**2-39**2 }{ 2 * 39 * 61 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1189.5 }{ 86.2 } = 13.8 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 61 }{ 2 * sin 57° 24'27" } = 36.2 ; ;
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