7 20 21 triangle

Acute scalene triangle.

Sides: a = 7   b = 20   c = 21

Area: T = 69.97114227381
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 19.46329509463° = 19°27'47″ = 0.34396925762 rad
Angle ∠ B = β = 72.17442941315° = 72°10'27″ = 1.26596790679 rad
Angle ∠ C = γ = 88.36327549222° = 88°21'46″ = 1.54222210095 rad

Height: ha = 19.9921835068
Height: hb = 6.99771422738
Height: hc = 6.66439450227

Median: ma = 20.20551973512
Median: mb = 12.04215945788
Median: mc = 10.68987791632

Inradius: r = 2.91554759474
Circumradius: R = 10.504428834

Vertex coordinates: A[21; 0] B[0; 0] C[2.14328571429; 6.66439450227]
Centroid: CG[7.71442857143; 2.22113150076]
Coordinates of the circumscribed circle: U[10.5; 0.3300122524]
Coordinates of the inscribed circle: I[4; 2.91554759474]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.5377049054° = 160°32'13″ = 0.34396925762 rad
∠ B' = β' = 107.8265705869° = 107°49'33″ = 1.26596790679 rad
∠ C' = γ' = 91.63772450778° = 91°38'14″ = 1.54222210095 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 7+20+21 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-7)(24-20)(24-21) } ; ; T = sqrt{ 4896 } = 69.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.97 }{ 7 } = 19.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.97 }{ 20 } = 7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.97 }{ 21 } = 6.66 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 19° 27'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-7**2-21**2 }{ 2 * 7 * 21 } ) = 72° 10'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-7**2-20**2 }{ 2 * 20 * 7 } ) = 88° 21'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.97 }{ 24 } = 2.92 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 19° 27'47" } = 10.5 ; ;




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