7 10 12 triangle

Acute scalene triangle.

Sides: a = 7 b = 10 c = 12

Area: T = 34.97876714491
Perimeter: p = 29
Semiperimeter: s = 14.5

Angle ∠ A = α = 35.65990876961° = 35°39'33″ = 0.62223684886 rad
Angle ∠ B = β = 56.38876254015° = 56°23'15″ = 0.98441497206 rad
Angle ∠ C = γ = 87.95332869023° = 87°57'12″ = 1.53550744444 rad

Height: h_a = 9.9943620414
Height: h_b = 6.99655342898
Height: h_c = 5.83296119082

Median: m_a = 10.47661634199
Median: m_b = 8.45657672626
Median: m_c = 6.2054836823

Inradius: r = 2.41222532034
Circumradius: R = 6.00438301951

Vertex coordinates: A[12; 0] B[0; 0] C[3.875; 5.83296119082]
Centroid: CG[5.29216666667; 1.94332039694]
Coordinates of the circumscribed circle: U[6; 0.2144422507]
Coordinates of the inscribed circle: I[4.5; 2.41222532034]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.3410912304° = 144°20'27″ = 0.62223684886 rad
∠ B' = β' = 123.6122374598° = 123°36'45″ = 0.98441497206 rad
∠ C' = γ' = 92.04767130977° = 92°2'48″ = 1.53550744444 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.

$a = 7 ; ; b = 10 ; ; c = 12 ; ;$

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

$p = a+b+c = 7+10+12 = 29 ; ;$

2. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 29 }{ 2 } = 14.5 ; ;$

3. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.5 * (14.5-7)(14.5-10)(14.5-12) } ; ; T = sqrt{ 1223.44 } = 34.98 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34.98 }{ 7 } = 9.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34.98 }{ 10 } = 7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34.98 }{ 12 } = 5.83 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 7**2-10**2-12**2 }{ 2 * 10 * 12 } ) = 35° 39'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-7**2-12**2 }{ 2 * 7 * 12 } ) = 56° 23'15" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-7**2-10**2 }{ 2 * 10 * 7 } ) = 87° 57'12" ; ;$

6. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 34.98 }{ 14.5 } = 2.41 ; ;$

7. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 35° 39'33" } = 6 ; ;$

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