Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and c (hypotenuse-calculated).

Right scalene triangle.

Sides: a = 5   b = 12   c = 13

Area: T = 30
Perimeter: p = 30
Semiperimeter: s = 15

Angle ∠ A = α = 22.6219864948° = 22°37'11″ = 0.39547911197 rad
Angle ∠ B = β = 67.3880135052° = 67°22'49″ = 1.17660052071 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 12
Height: hb = 5
Height: hc = 4.61553846154

Median: ma = 12.25876506721
Median: mb = 7.81102496759
Median: mc = 6.5

Inradius: r = 2
Circumradius: R = 6.5

Vertex coordinates: A[13; 0] B[0; 0] C[1.92330769231; 4.61553846154]
Centroid: CG[4.97443589744; 1.53884615385]
Coordinates of the circumscribed circle: U[6.5; -0]
Coordinates of the inscribed circle: I[3; 2]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.3880135052° = 157°22'49″ = 0.39547911197 rad
∠ B' = β' = 112.6219864948° = 112°37'11″ = 1.17660052071 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and c (hypotenuse-calculated).

a = 5 ; ; b = 12 ; ; c = 13 ; ;


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 ; ; b = 12 ; ; c = 13 ; ; : Nr. 1

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

p = a+b+c = 5+12+13 = 30 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30 }{ 2 } = 15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15 * (15-5)(15-12)(15-13) } ; ; T = sqrt{ 900 } = 30 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 30 }{ 5 } = 12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 30 }{ 12 } = 5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 30 }{ 13 } = 4.62 ; ;

6. 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12**2+13**2-5**2 }{ 2 * 12 * 13 } ) = 22° 37'11" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+13**2-12**2 }{ 2 * 5 * 13 } ) = 67° 22'49" ; ; gamma = 180° - alpha - beta = 180° - 22° 37'11" - 67° 22'49" = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 30 }{ 15 } = 2 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5 }{ 2 * sin 22° 37'11" } = 6.5 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 13**2 - 5**2 } }{ 2 } = 12.258 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 5**2 - 12**2 } }{ 2 } = 7.81 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 5**2 - 13**2 } }{ 2 } = 6.5 ; ;
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