Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 66.175   b = 66.175   c = 25.82202041188

Area: T = 837.9110369175
Perimeter: p = 158.1770204119
Semiperimeter: s = 79.08551020594

Angle ∠ A = α = 78.75° = 78°45' = 1.37444467859 rad
Angle ∠ B = β = 78.75° = 78°45' = 1.37444467859 rad
Angle ∠ C = γ = 22.5° = 22°30' = 0.39326990817 rad

Height: ha = 25.32440761368
Height: hb = 25.32440761368
Height: hc = 64.90334659307

Median: ma = 37.7910529589
Median: mb = 37.7910529589
Median: mc = 64.90334659307

Inradius: r = 10.59550469476
Circumradius: R = 33.73657224472

Vertex coordinates: A[25.82202041188; 0] B[0; 0] C[12.91101020594; 64.90334659307]
Centroid: CG[12.91101020594; 21.63444886436]
Coordinates of the circumscribed circle: U[12.91101020594; 31.16877434835]
Coordinates of the inscribed circle: I[12.91101020594; 10.59550469476]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 101.25° = 101°15' = 1.37444467859 rad
∠ B' = β' = 101.25° = 101°15' = 1.37444467859 rad
∠ C' = γ' = 157.5° = 157°30' = 0.39326990817 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 = 66.18 ; ; b = 66.18 ; ; gamma = 22° 30' ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 66.18**2+66.18**2 - 2 * 66.18 * 66.18 * cos(22° 30') } ; ; c = 25.82 ; ;


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 = 66.18 ; ; b = 66.18 ; ; c = 25.82 ; ;

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

p = a+b+c = 66.18+66.18+25.82 = 158.17 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 158.17 }{ 2 } = 79.09 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 79.09 * (79.09-66.18)(79.09-66.18)(79.09-25.82) } ; ; T = sqrt{ 702093.79 } = 837.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 837.91 }{ 66.18 } = 25.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 837.91 }{ 66.18 } = 25.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 837.91 }{ 25.82 } = 64.9 ; ;

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{ 66.18**2-66.18**2-25.82**2 }{ 2 * 66.18 * 25.82 } ) = 78° 45' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 66.18**2-66.18**2-25.82**2 }{ 2 * 66.18 * 25.82 } ) = 78° 45' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25.82**2-66.18**2-66.18**2 }{ 2 * 66.18 * 66.18 } ) = 22° 30' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 837.91 }{ 79.09 } = 10.6 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 66.18 }{ 2 * sin 78° 45' } = 33.74 ; ;




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