12 12 21 triangle

Obtuse isosceles triangle.

Sides: a = 12   b = 12   c = 21

Area: T = 60.99994877028
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 28.95550243719° = 28°57'18″ = 0.50553605103 rad
Angle ∠ B = β = 28.95550243719° = 28°57'18″ = 0.50553605103 rad
Angle ∠ C = γ = 122.0989951256° = 122°5'24″ = 2.1310871633 rad

Height: ha = 10.16765812838
Height: hb = 10.16765812838
Height: hc = 5.80994750193

Median: ma = 16.0165617378
Median: mb = 16.0165617378
Median: mc = 5.80994750193

Inradius: r = 2.71110883423
Circumradius: R = 12.39435467079

Vertex coordinates: A[21; 0] B[0; 0] C[10.5; 5.80994750193]
Centroid: CG[10.5; 1.93664916731]
Coordinates of the circumscribed circle: U[10.5; -6.58440716886]
Coordinates of the inscribed circle: I[10.5; 2.71110883423]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.0454975628° = 151°2'42″ = 0.50553605103 rad
∠ B' = β' = 151.0454975628° = 151°2'42″ = 0.50553605103 rad
∠ C' = γ' = 57.91100487437° = 57°54'36″ = 2.1310871633 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 = 12 ; ; c = 21 ; ;

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

p = a+b+c = 12+12+21 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-12)(22.5-12)(22.5-21) } ; ; T = sqrt{ 3720.94 } = 61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 61 }{ 12 } = 10.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 61 }{ 12 } = 10.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 61 }{ 21 } = 5.81 ; ;

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{ 12**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 28° 57'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 28° 57'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 122° 5'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 61 }{ 22.5 } = 2.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 28° 57'18" } = 12.39 ; ;




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

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