Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 78   b = 78   c = 65.92884488315

Area: T = 2330.307719597
Perimeter: p = 221.9288448831
Semiperimeter: s = 110.9644224416

Angle ∠ A = α = 65° = 1.13444640138 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 59.75114665633
Height: hb = 59.75114665633
Height: hc = 70.69220073889

Median: ma = 60.78105905094
Median: mb = 60.78105905094
Median: mc = 70.69220073889

Inradius: r = 21.00105270459
Circumradius: R = 43.03217388395

Vertex coordinates: A[65.92884488315; 0] B[0; 0] C[32.96442244158; 70.69220073889]
Centroid: CG[32.96442244158; 23.5644002463]
Coordinates of the circumscribed circle: U[32.96442244158; 27.66602685493]
Coordinates of the inscribed circle: I[32.96442244158; 21.00105270459]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115° = 1.13444640138 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 130° = 0.8732664626 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 = 78 ; ; b = 78 ; ; gamma = 50° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 78**2+78**2 - 2 * 78 * 78 * cos(50° ) } ; ; c = 65.93 ; ;


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 = 78 ; ; b = 78 ; ; c = 65.93 ; ;

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

p = a+b+c = 78+78+65.93 = 221.93 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 221.93 }{ 2 } = 110.96 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 110.96 * (110.96-78)(110.96-78)(110.96-65.93) } ; ; T = sqrt{ 5430331.63 } = 2330.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2330.31 }{ 78 } = 59.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2330.31 }{ 78 } = 59.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2330.31 }{ 65.93 } = 70.69 ; ;

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{ 78**2-78**2-65.93**2 }{ 2 * 78 * 65.93 } ) = 65° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 78**2-78**2-65.93**2 }{ 2 * 78 * 65.93 } ) = 65° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 65.93**2-78**2-78**2 }{ 2 * 78 * 78 } ) = 50° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2330.31 }{ 110.96 } = 21 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 78 }{ 2 * sin 65° } = 43.03 ; ;




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