Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 75   b = 75   c = 106.0666017178

Area: T = 2812.5
Perimeter: p = 256.0666017178
Semiperimeter: s = 128.0333008589

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 75
Height: hb = 75
Height: hc = 53.0333008589

Median: ma = 83.85325491562
Median: mb = 83.85325491562
Median: mc = 53.0333008589

Inradius: r = 21.9676991411
Circumradius: R = 53.0333008589

Vertex coordinates: A[106.0666017178; 0] B[0; 0] C[53.0333008589; 53.0333008589]
Centroid: CG[53.0333008589; 17.67876695297]
Coordinates of the circumscribed circle: U[53.0333008589; 0]
Coordinates of the inscribed circle: I[53.0333008589; 21.9676991411]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 75 ; ; b = 75 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 75**2+75**2 - 2 * 75 * 75 * cos(90° ) } ; ; c = 106.07 ; ;


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 = 75 ; ; b = 75 ; ; c = 106.07 ; ;

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

p = a+b+c = 75+75+106.07 = 256.07 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 256.07 }{ 2 } = 128.03 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 128.03 * (128.03-75)(128.03-75)(128.03-106.07) } ; ; T = sqrt{ 7910156.25 } = 2812.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2812.5 }{ 75 } = 75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2812.5 }{ 75 } = 75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2812.5 }{ 106.07 } = 53.03 ; ;

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{ 75**2-75**2-106.07**2 }{ 2 * 75 * 106.07 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 75**2-75**2-106.07**2 }{ 2 * 75 * 106.07 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 106.07**2-75**2-75**2 }{ 2 * 75 * 75 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2812.5 }{ 128.03 } = 21.97 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 75 }{ 2 * sin 45° } = 53.03 ; ;

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