13 19 21 triangle

Acute scalene triangle.

Sides: a = 13   b = 19   c = 21

Area: T = 121.479916488
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 37.51113170202° = 37°30'41″ = 0.65546959888 rad
Angle ∠ B = β = 62.86878756647° = 62°52'4″ = 1.09772514241 rad
Angle ∠ C = γ = 79.62108073151° = 79°37'15″ = 1.39896452407 rad

Height: ha = 18.68991022893
Height: hb = 12.78772805137
Height: hc = 11.56994442743

Median: ma = 18.94106969249
Median: mb = 14.65443508898
Median: mc = 12.44398553046

Inradius: r = 4.58441194294
Circumradius: R = 10.67546700249

Vertex coordinates: A[21; 0] B[0; 0] C[5.92985714286; 11.56994442743]
Centroid: CG[8.97661904762; 3.85664814248]
Coordinates of the circumscribed circle: U[10.5; 1.9233169296]
Coordinates of the inscribed circle: I[7.5; 4.58441194294]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.489868298° = 142°29'19″ = 0.65546959888 rad
∠ B' = β' = 117.1322124335° = 117°7'56″ = 1.09772514241 rad
∠ C' = γ' = 100.3799192685° = 100°22'45″ = 1.39896452407 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 = 13 ; ; b = 19 ; ; c = 21 ; ;

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

p = a+b+c = 13+19+21 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-13)(26.5-19)(26.5-21) } ; ; T = sqrt{ 14757.19 } = 121.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 121.48 }{ 13 } = 18.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 121.48 }{ 19 } = 12.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 121.48 }{ 21 } = 11.57 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 13**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 37° 30'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 62° 52'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-13**2-19**2 }{ 2 * 19 * 13 } ) = 79° 37'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 121.48 }{ 26.5 } = 4.58 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 37° 30'41" } = 10.67 ; ;




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

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