Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Acute isosceles triangle.

Sides: a = 200   b = 200   c = 103.5287618041

Area: T = 10000
Perimeter: p = 503.5287618041
Semiperimeter: s = 251.764380902

Angle ∠ A = α = 75° = 1.3098996939 rad
Angle ∠ B = β = 75° = 1.3098996939 rad
Angle ∠ C = γ = 30° = 0.52435987756 rad

Height: ha = 100
Height: hb = 100
Height: hc = 193.1855165258

Median: ma = 123.9311367493
Median: mb = 123.9311367493
Median: mc = 193.1855165258

Inradius: r = 39.7219767662
Circumradius: R = 103.5287618041

Vertex coordinates: A[103.5287618041; 0] B[0; 0] C[51.76438090205; 193.1855165258]
Centroid: CG[51.76438090205; 64.39550550859]
Coordinates of the circumscribed circle: U[51.76438090205; 89.65875472168]
Coordinates of the inscribed circle: I[51.76438090205; 39.7219767662]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105° = 1.3098996939 rad
∠ B' = β' = 105° = 1.3098996939 rad
∠ C' = γ' = 150° = 0.52435987756 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

a = 200 ; ; b = 200 ; ; gamma = 30° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 200**2+200**2 - 2 * 200 * 200 * cos(30° ) } ; ; c = 103.53 ; ;


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 = 200 ; ; b = 200 ; ; c = 103.53 ; ;

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

p = a+b+c = 200+200+103.53 = 503.53 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 503.53 }{ 2 } = 251.76 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 251.76 * (251.76-200)(251.76-200)(251.76-103.53) } ; ; T = sqrt{ 100000000 } = 10000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10000 }{ 200 } = 100 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10000 }{ 200 } = 100 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10000 }{ 103.53 } = 193.19 ; ;

6. 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{ 200**2-200**2-103.53**2 }{ 2 * 200 * 103.53 } ) = 75° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 200**2-200**2-103.53**2 }{ 2 * 200 * 103.53 } ) = 75° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 103.53**2-200**2-200**2 }{ 2 * 200 * 200 } ) = 30° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10000 }{ 251.76 } = 39.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 200 }{ 2 * sin 75° } = 103.53 ; ;




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