Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, c and angle β.

Acute isosceles triangle.

Sides: a = 50   b = 21.64439613938   c = 50

Area: T = 528.2732827176
Perimeter: p = 121.6443961394
Semiperimeter: s = 60.82219806969

Angle ∠ A = α = 77.5° = 77°30' = 1.35326301703 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 77.5° = 77°30' = 1.35326301703 rad

Height: ha = 21.1310913087
Height: hb = 48.8154800356
Height: hc = 21.1310913087

Median: ma = 29.31326343478
Median: mb = 48.8154800356
Median: mc = 29.31326343478

Inradius: r = 8.68655577724
Circumradius: R = 25.60769878579

Vertex coordinates: A[50; 0] B[0; 0] C[45.31553893518; 21.1310913087]
Centroid: CG[31.77217964506; 7.04436376957]
Coordinates of the circumscribed circle: U[25; 5.54223665661]
Coordinates of the inscribed circle: I[39.17880193031; 8.68655577724]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102.5° = 102°30' = 1.35326301703 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 102.5° = 102°30' = 1.35326301703 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle β.

a = 50 ; ; c = 50 ; ; beta = 25° ; ;

2. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 50**2+50**2 - 2 * 50 * 50 * cos(25° ) } ; ; b = 21.64 ; ;


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 = 50 ; ; b = 21.64 ; ; c = 50 ; ;

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

p = a+b+c = 50+21.64+50 = 121.64 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 121.64 }{ 2 } = 60.82 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60.82 * (60.82-50)(60.82-21.64)(60.82-50) } ; ; T = sqrt{ 279072.18 } = 528.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 528.27 }{ 50 } = 21.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 528.27 }{ 21.64 } = 48.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 528.27 }{ 50 } = 21.13 ; ;

7. 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{ 50**2-21.64**2-50**2 }{ 2 * 21.64 * 50 } ) = 77° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21.64**2-50**2-50**2 }{ 2 * 50 * 50 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 50**2-50**2-21.64**2 }{ 2 * 21.64 * 50 } ) = 77° 30' ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 528.27 }{ 60.82 } = 8.69 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 77° 30' } = 25.61 ; ;




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