9 10 12 triangle

Acute scalene triangle.

Sides: a = 9   b = 10   c = 12

Area: T = 44.03990451758
Perimeter: p = 31
Semiperimeter: s = 15.5

Angle ∠ A = α = 47.22114422911° = 47°13'17″ = 0.82441696455 rad
Angle ∠ B = β = 54.64105803778° = 54°38'26″ = 0.95436580328 rad
Angle ∠ C = γ = 78.13879773311° = 78°8'17″ = 1.36437649753 rad

Height: ha = 9.78664544835
Height: hb = 8.80878090352
Height: hc = 7.34398408626

Median: ma = 10.08771205009
Median: mb = 9.35441434669
Median: mc = 7.38224115301

Inradius: r = 2.8411228721
Circumradius: R = 6.13109231143

Vertex coordinates: A[12; 0] B[0; 0] C[5.20883333333; 7.34398408626]
Centroid: CG[5.73661111111; 2.44766136209]
Coordinates of the circumscribed circle: U[6; 1.26602453068]
Coordinates of the inscribed circle: I[5.5; 2.8411228721]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.7798557709° = 132°46'43″ = 0.82441696455 rad
∠ B' = β' = 125.3599419622° = 125°21'34″ = 0.95436580328 rad
∠ C' = γ' = 101.8622022669° = 101°51'43″ = 1.36437649753 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 = 9 ; ; b = 10 ; ; c = 12 ; ;

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

p = a+b+c = 9+10+12 = 31 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31 }{ 2 } = 15.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.5 * (15.5-9)(15.5-10)(15.5-12) } ; ; T = sqrt{ 1939.44 } = 44.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44.04 }{ 9 } = 9.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44.04 }{ 10 } = 8.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44.04 }{ 12 } = 7.34 ; ;

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{ 9**2-10**2-12**2 }{ 2 * 10 * 12 } ) = 47° 13'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-9**2-12**2 }{ 2 * 9 * 12 } ) = 54° 38'26" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-9**2-10**2 }{ 2 * 10 * 9 } ) = 78° 8'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44.04 }{ 15.5 } = 2.84 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 47° 13'17" } = 6.13 ; ;




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