13 20 28 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 20   c = 28

Area: T = 118.3687806012
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 25.00878332347° = 25°28″ = 0.43664690287 rad
Angle ∠ B = β = 40.57696972304° = 40°34'11″ = 0.70880747932 rad
Angle ∠ C = γ = 114.4222469535° = 114°25'21″ = 1.99770488316 rad

Height: ha = 18.21104316941
Height: hb = 11.83767806012
Height: hc = 8.45548432865

Median: ma = 23.44767481754
Median: mb = 19.40436079119
Median: mc = 9.40774438611

Inradius: r = 3.88109116725
Circumradius: R = 15.37658024359

Vertex coordinates: A[28; 0] B[0; 0] C[9.875; 8.45548432865]
Centroid: CG[12.625; 2.81882810955]
Coordinates of the circumscribed circle: U[14; -6.35773029302]
Coordinates of the inscribed circle: I[10.5; 3.88109116725]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.9922166765° = 154°59'32″ = 0.43664690287 rad
∠ B' = β' = 139.433030277° = 139°25'49″ = 0.70880747932 rad
∠ C' = γ' = 65.5787530465° = 65°34'39″ = 1.99770488316 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 = 13 ; ; b = 20 ; ; c = 28 ; ;

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

p = a+b+c = 13+20+28 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-13)(30.5-20)(30.5-28) } ; ; T = sqrt{ 14010.94 } = 118.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 118.37 }{ 13 } = 18.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 118.37 }{ 20 } = 11.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 118.37 }{ 28 } = 8.45 ; ;

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{ 13**2-20**2-28**2 }{ 2 * 20 * 28 } ) = 25° 28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-13**2-28**2 }{ 2 * 13 * 28 } ) = 40° 34'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-13**2-20**2 }{ 2 * 20 * 13 } ) = 114° 25'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 118.37 }{ 30.5 } = 3.88 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 25° 28" } = 15.38 ; ;




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