12 14 21 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 14   c = 21

Area: T = 80.11551515008
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 33.02547340605° = 33°1'29″ = 0.5766390344 rad
Angle ∠ B = β = 39.48219079378° = 39°28'55″ = 0.68990892885 rad
Angle ∠ C = γ = 107.4933358002° = 107°29'36″ = 1.87661130212 rad

Height: ha = 13.35325252501
Height: hb = 11.4455021643
Height: hc = 7.63300144286

Median: ma = 16.8087736314
Median: mb = 15.60444865343
Median: mc = 7.73298124169

Inradius: r = 3.4099155383
Circumradius: R = 11.00991534932

Vertex coordinates: A[21; 0] B[0; 0] C[9.26219047619; 7.63300144286]
Centroid: CG[10.08773015873; 2.54333381429]
Coordinates of the circumscribed circle: U[10.5; -3.30992991155]
Coordinates of the inscribed circle: I[9.5; 3.4099155383]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.9755265939° = 146°58'31″ = 0.5766390344 rad
∠ B' = β' = 140.5188092062° = 140°31'5″ = 0.68990892885 rad
∠ C' = γ' = 72.50766419983° = 72°30'24″ = 1.87661130212 rad

Calculate another triangle




How did we calculate this triangle?

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 ; ; b = 14 ; ; c = 21 ; ;

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

p = a+b+c = 12+14+21 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-12)(23.5-14)(23.5-21) } ; ; T = sqrt{ 6418.44 } = 80.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80.12 }{ 12 } = 13.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80.12 }{ 14 } = 11.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80.12 }{ 21 } = 7.63 ; ;

5. 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{ 14**2+21**2-12**2 }{ 2 * 14 * 21 } ) = 33° 1'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+21**2-14**2 }{ 2 * 12 * 21 } ) = 39° 28'55" ; ; gamma = 180° - alpha - beta = 180° - 33° 1'29" - 39° 28'55" = 107° 29'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80.12 }{ 23.5 } = 3.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 33° 1'29" } = 11.01 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 21**2 - 12**2 } }{ 2 } = 16.808 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 12**2 - 14**2 } }{ 2 } = 15.604 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 12**2 - 21**2 } }{ 2 } = 7.73 ; ;
Calculate another triangle

Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.