Isosceles triangle calculator (a)

Please enter two properties of the isosceles triangle

Use symbols: a, b, h, T, p, A, B, C, r, R


You have entered side a and perimeter p.

Acute isosceles triangle.

Sides: a = 76   b = 76   c = 99

Area: T = 2854.631113861
Perimeter: p = 251
Semiperimeter: s = 125.5

Angle ∠ A = α = 49.35991196256° = 49°21'33″ = 0.86114791534 rad
Angle ∠ B = β = 49.35991196256° = 49°21'33″ = 0.86114791534 rad
Angle ∠ C = γ = 81.28217607489° = 81°16'54″ = 1.41986343469 rad

Height: ha = 75.12218720686
Height: hb = 75.12218720686
Height: hc = 57.66993159314

Median: ma = 79.65223697074
Median: mb = 79.65223697074
Median: mc = 57.66993159314

Inradius: r = 22.74660648495
Circumradius: R = 50.0798624193

Vertex coordinates: A[99; 0] B[0; 0] C[49.5; 57.66993159314]
Centroid: CG[49.5; 19.22331053105]
Coordinates of the circumscribed circle: U[49.5; 7.59106917384]
Coordinates of the inscribed circle: I[49.5; 22.74660648495]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.6410880374° = 130°38'27″ = 0.86114791534 rad
∠ B' = β' = 130.6410880374° = 130°38'27″ = 0.86114791534 rad
∠ C' = γ' = 98.71882392511° = 98°43'6″ = 1.41986343469 rad

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How did we calculate this triangle?

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 = 76 ; ; b = 76 ; ; c = 99 ; ;

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

p = a+b+c = 76+76+99 = 251 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 251 }{ 2 } = 125.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 125.5 * (125.5-76)(125.5-76)(125.5-99) } ; ; T = sqrt{ 8148918.94 } = 2854.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2854.63 }{ 76 } = 75.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2854.63 }{ 76 } = 75.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2854.63 }{ 99 } = 57.67 ; ;

5. 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{ 76**2-76**2-99**2 }{ 2 * 76 * 99 } ) = 49° 21'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 76**2-76**2-99**2 }{ 2 * 76 * 99 } ) = 49° 21'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 99**2-76**2-76**2 }{ 2 * 76 * 76 } ) = 81° 16'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2854.63 }{ 125.5 } = 22.75 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 76 }{ 2 * sin 49° 21'33" } = 50.08 ; ;




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