7 8 10 triangle

Acute scalene triangle.

Sides: a = 7   b = 8   c = 10

Area: T = 27.81107443266
Perimeter: p = 25
Semiperimeter: s = 12.5

Angle ∠ A = α = 44.04986256741° = 44°2'55″ = 0.7698793549 rad
Angle ∠ B = β = 52.61768015821° = 52°37' = 0.91883364295 rad
Angle ∠ C = γ = 83.33545727438° = 83°20'4″ = 1.45444626751 rad

Height: ha = 7.94659269505
Height: hb = 6.95326860817
Height: hc = 5.56221488653

Median: ma = 8.35216465442
Median: mb = 7.64985292704
Median: mc = 5.61224860802

Inradius: r = 2.22548595461
Circumradius: R = 5.03440256397

Vertex coordinates: A[10; 0] B[0; 0] C[4.25; 5.56221488653]
Centroid: CG[4.75; 1.85440496218]
Coordinates of the circumscribed circle: U[5; 0.58443065475]
Coordinates of the inscribed circle: I[4.5; 2.22548595461]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.9511374326° = 135°57'5″ = 0.7698793549 rad
∠ B' = β' = 127.3833198418° = 127°23' = 0.91883364295 rad
∠ C' = γ' = 96.66554272562° = 96°39'56″ = 1.45444626751 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 = 8 ; ; c = 10 ; ;

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

p = a+b+c = 7+8+10 = 25 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25 }{ 2 } = 12.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.5 * (12.5-7)(12.5-8)(12.5-10) } ; ; T = sqrt{ 773.44 } = 27.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27.81 }{ 7 } = 7.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27.81 }{ 8 } = 6.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27.81 }{ 10 } = 5.56 ; ;

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**2-8**2-10**2 }{ 2 * 8 * 10 } ) = 44° 2'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8**2-7**2-10**2 }{ 2 * 7 * 10 } ) = 52° 37' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10**2-7**2-8**2 }{ 2 * 8 * 7 } ) = 83° 20'4" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27.81 }{ 12.5 } = 2.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 44° 2'55" } = 5.03 ; ;




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