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 = 164   b = 68   c = 177.5398728169

Area: T = 5576
Perimeter: p = 409.5398728169
Semiperimeter: s = 204.7699364085

Angle ∠ A = α = 67.47994343971° = 67°28'46″ = 1.17877383076 rad
Angle ∠ B = β = 22.52105656029° = 22°31'14″ = 0.39330580192 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 68
Height: hb = 164
Height: hc = 62.81444637229

Median: ma = 106.5276991885
Median: mb = 167.4877312952
Median: mc = 88.76993640847

Inradius: r = 27.23106359153
Circumradius: R = 88.76993640847

Vertex coordinates: A[177.5398728169; 0] B[0; 0] C[151.4943706626; 62.81444637229]
Centroid: CG[109.6777478265; 20.93881545743]
Coordinates of the circumscribed circle: U[88.76993640847; 0]
Coordinates of the inscribed circle: I[136.7699364085; 27.23106359153]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5210565603° = 112°31'14″ = 1.17877383076 rad
∠ B' = β' = 157.4799434397° = 157°28'46″ = 0.39330580192 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 = 164 ; ; b = 68 ; ;

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{ 164**2 + 68**2 } = 177.539 ; ;


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 = 164 ; ; b = 68 ; ; c = 177.54 ; ;

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

p = a+b+c = 164+68+177.54 = 409.54 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 409.54 }{ 2 } = 204.77 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 204.77 * (204.77-164)(204.77-68)(204.77-177.54) } ; ; T = sqrt{ 31091776 } = 5576 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5576 }{ 164 } = 68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5576 }{ 68 } = 164 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5576 }{ 177.54 } = 62.81 ; ;

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{ 164**2-68**2-177.54**2 }{ 2 * 68 * 177.54 } ) = 67° 28'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 68**2-164**2-177.54**2 }{ 2 * 164 * 177.54 } ) = 22° 31'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 177.54**2-164**2-68**2 }{ 2 * 68 * 164 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5576 }{ 204.77 } = 27.23 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 164 }{ 2 * sin 67° 28'46" } = 88.77 ; ;
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