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 = 12.5   b = 3.5   c = 12.98107549857

Area: T = 21.875
Perimeter: p = 28.98107549857
Semiperimeter: s = 14.49903774929

Angle ∠ A = α = 74.35877535428° = 74°21'28″ = 1.29877876237 rad
Angle ∠ B = β = 15.64222464572° = 15°38'32″ = 0.27330087031 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 3.5
Height: hb = 12.5
Height: hc = 3.3770374069

Median: ma = 7.16332743909
Median: mb = 12.62219055614
Median: mc = 6.49903774929

Inradius: r = 1.51096225071
Circumradius: R = 6.49903774929

Vertex coordinates: A[12.98107549857; 0] B[0; 0] C[12.03770502464; 3.3770374069]
Centroid: CG[8.33992684107; 1.1233458023]
Coordinates of the circumscribed circle: U[6.49903774929; 0]
Coordinates of the inscribed circle: I[10.99903774929; 1.51096225071]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.6422246457° = 105°38'32″ = 1.29877876237 rad
∠ B' = β' = 164.3587753543° = 164°21'28″ = 0.27330087031 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 = 12.5 ; ; b = 3.5 ; ;

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{ 12.5**2 + 3.5**2 } = 12.981 ; ;


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 = 12.5 ; ; b = 3.5 ; ; c = 12.98 ; ;

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

p = a+b+c = 12.5+3.5+12.98 = 28.98 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28.98 }{ 2 } = 14.49 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.49 * (14.49-12.5)(14.49-3.5)(14.49-12.98) } ; ; T = sqrt{ 478.52 } = 21.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.88 }{ 12.5 } = 3.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.88 }{ 3.5 } = 12.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.88 }{ 12.98 } = 3.37 ; ;

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{ 12.5**2-3.5**2-12.98**2 }{ 2 * 3.5 * 12.98 } ) = 74° 21'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.5**2-12.5**2-12.98**2 }{ 2 * 12.5 * 12.98 } ) = 15° 38'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.98**2-12.5**2-3.5**2 }{ 2 * 3.5 * 12.5 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.88 }{ 14.49 } = 1.51 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.5 }{ 2 * sin 74° 21'28" } = 6.49 ; ;
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