Triangle calculator

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

You have entered side b, c and angle α.

Acute isosceles triangle.

Sides: a = 34.44215089129   b = 45   c = 45

Area: T = 715.9465615951
Perimeter: p = 124.4421508913
Semiperimeter: s = 62.22107544564

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 67.5° = 67°30' = 1.17880972451 rad

Height: ha = 41.5754578963
Height: hb = 31.82198051534
Height: hc = 31.82198051534

Median: ma = 41.5754578963
Median: mb = 33.15765795597
Median: mc = 33.15765795597

Inradius: r = 11.50765402566
Circumradius: R = 24.35438245066

Vertex coordinates: A[45; 0] B[0; 0] C[13.18801948466; 31.82198051534]
Centroid: CG[19.39333982822; 10.60766017178]
Coordinates of the circumscribed circle: U[22.5; 9.32198051534]
Coordinates of the inscribed circle: I[17.22107544564; 11.50765402566]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 rad
∠ C' = γ' = 112.5° = 112°30' = 1.17880972451 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 45 ; ; c = 45 ; ; alpha = 45° ; ;

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

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 45**2+45**2 - 2 * 45 * 45 * cos 45° } ; ; a = 34.44 ; ;


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 = 34.44 ; ; b = 45 ; ; c = 45 ; ;

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

p = a+b+c = 34.44+45+45 = 124.44 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 124.44 }{ 2 } = 62.22 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.22 * (62.22-34.44)(62.22-45)(62.22-45) } ; ; T = sqrt{ 512578.13 } = 715.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 715.95 }{ 34.44 } = 41.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 715.95 }{ 45 } = 31.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 715.95 }{ 45 } = 31.82 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 45**2+45**2-34.44**2 }{ 2 * 45 * 45 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 34.44**2+45**2-45**2 }{ 2 * 34.44 * 45 } ) = 67° 30' ; ; gamma = 180° - alpha - beta = 180° - 45° - 67° 30' = 67° 30' ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 715.95 }{ 62.22 } = 11.51 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 34.44 }{ 2 * sin 45° } = 24.35 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 45**2 - 34.44**2 } }{ 2 } = 41.575 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 34.44**2 - 45**2 } }{ 2 } = 33.157 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 34.44**2 - 45**2 } }{ 2 } = 33.157 ; ;
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