Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 45   b = 45   c = 63.64396103068

Area: T = 1012.5
Perimeter: p = 153.6439610307
Semiperimeter: s = 76.82198051534

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

Height: ha = 45
Height: hb = 45
Height: hc = 31.82198051534

Median: ma = 50.31215294937
Median: mb = 50.31215294937
Median: mc = 31.82198051534

Inradius: r = 13.18801948466
Circumradius: R = 31.82198051534

Vertex coordinates: A[63.64396103068; 0] B[0; 0] C[31.82198051534; 31.82198051534]
Centroid: CG[31.82198051534; 10.60766017178]
Coordinates of the circumscribed circle: U[31.82198051534; 0]
Coordinates of the inscribed circle: I[31.82198051534; 13.18801948466]

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

Calculate another triangle




How did we calculate this triangle?

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

a = 45 ; ; b = 45 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 45**2+45**2 - 2 * 45 * 45 * cos(90° ) } ; ; c = 63.64 ; ;


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 = 45 ; ; b = 45 ; ; c = 63.64 ; ;

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

p = a+b+c = 45+45+63.64 = 153.64 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 153.64 }{ 2 } = 76.82 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 76.82 * (76.82-45)(76.82-45)(76.82-63.64) } ; ; T = sqrt{ 1025156.25 } = 1012.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1012.5 }{ 45 } = 45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1012.5 }{ 45 } = 45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1012.5 }{ 63.64 } = 31.82 ; ;

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{ 45**2-45**2-63.64**2 }{ 2 * 45 * 63.64 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-45**2-63.64**2 }{ 2 * 45 * 63.64 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 63.64**2-45**2-45**2 }{ 2 * 45 * 45 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1012.5 }{ 76.82 } = 13.18 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45 }{ 2 * sin 45° } = 31.82 ; ;




Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.