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: ha = 9.9943620414
Height: hb = 6.99655342898
Height: hc = 5.83296119082

Median: ma = 10.47661634199
Median: mb = 8.45657672626
Median: mc = 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

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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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 10**2+12**2-7**2 }{ 2 * 10 * 12 } ) = 35° 39'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7**2+12**2-10**2 }{ 2 * 7 * 12 } ) = 56° 23'15" ; ; gamma = 180° - alpha - beta = 180° - 35° 39'33" - 56° 23'15" = 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 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10**2+2 * 12**2 - 7**2 } }{ 2 } = 10.476 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 7**2 - 10**2 } }{ 2 } = 8.456 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10**2+2 * 7**2 - 12**2 } }{ 2 } = 6.205 ; ;
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