12 21 27 triangle

Obtuse scalene triangle.

Sides: a = 12 b = 21 c = 27

Area: T = 120.7487670785
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 25.20987652968° = 25°12'32″ = 0.44399759548 rad
Angle ∠ B = β = 48.19896851042° = 48°11'23″ = 0.84110686706 rad
Angle ∠ C = γ = 106.6021549599° = 106°36'6″ = 1.86105480282 rad

Height: h_a = 20.12546117975
Height: h_b = 11.549977817
Height: h_c = 8.944427191

Median: m_a = 23.43107490277
Median: m_b = 18.06223918682
Median: m_c = 10.5

Inradius: r = 4.02549223595
Circumradius: R = 14.08772282582

Vertex coordinates: A[27; 0] B[0; 0] C[8; 8.944427191]
Centroid: CG[11.66766666667; 2.981142397]
Coordinates of the circumscribed circle: U[13.5; -4.02549223595]
Coordinates of the inscribed circle: I[9; 4.02549223595]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.7911234703° = 154°47'28″ = 0.44399759548 rad
∠ B' = β' = 131.8110314896° = 131°48'37″ = 0.84110686706 rad
∠ C' = γ' = 73.3988450401° = 73°23'54″ = 1.86105480282 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.

$a = 12 ; ; b = 21 ; ; c = 27 ; ;$

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

$p = a+b+c = 12+21+27 = 60 ; ;$

2. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;$

3. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-12)(30-21)(30-27) } ; ; T = sqrt{ 14580 } = 120.75 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 120.75 }{ 12 } = 20.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 120.75 }{ 21 } = 11.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 120.75 }{ 27 } = 8.94 ; ;$

5. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 12**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 25° 12'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 48° 11'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-21**2 }{ 2 * 21 * 12 } ) = 106° 36'6" ; ;$

6. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 120.75 }{ 30 } = 4.02 ; ;$

7. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 25° 12'32" } = 14.09 ; ;$

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

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