Triangle calculator

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

You have entered side a, c and area S.

Triangle has two solutions: a=90; b=158.1211068912; c=90 and a=90; b=86.01100434027; c=90.

#1 Obtuse isosceles triangle.

Sides: a = 90   b = 158.1211068912   c = 90

Area: T = 3400
Perimeter: p = 338.1211068912
Semiperimeter: s = 169.0610534456

Angle ∠ A = α = 28.54440046877° = 28°32'38″ = 0.49881868635 rad
Angle ∠ B = β = 122.9121990625° = 122°54'43″ = 2.14552189266 rad
Angle ∠ C = γ = 28.54440046877° = 28°32'38″ = 0.49881868635 rad

Height: ha = 75.55655555556
Height: hb = 43.00550217013
Height: hc = 75.55655555556

Median: ma = 120.5244421662
Median: mb = 43.00550217013
Median: mc = 120.5244421662

Inradius: r = 20.1111139545
Circumradius: R = 94.17550483961

Vertex coordinates: A[90; 0] B[0; 0] C[-48.90215135215; 75.55655555556]
Centroid: CG[13.69994954928; 25.18551851852]
Coordinates of the circumscribed circle: U[45; 82.72881073179]
Coordinates of the inscribed circle: I[10.9399465544; 20.1111139545]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.4565995312° = 151°27'22″ = 0.49881868635 rad
∠ B' = β' = 57.08880093755° = 57°5'17″ = 2.14552189266 rad
∠ C' = γ' = 151.4565995312° = 151°27'22″ = 0.49881868635 rad




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 = 90 ; ; b = 158.12 ; ; c = 90 ; ;

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

p = a+b+c = 90+158.12+90 = 338.12 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 338.12 }{ 2 } = 169.06 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 169.06 * (169.06-90)(169.06-158.12)(169.06-90) } ; ; T = sqrt{ 11560000 } = 3400 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3400 }{ 90 } = 75.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3400 }{ 158.12 } = 43.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3400 }{ 90 } = 75.56 ; ;

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{ 90**2-158.12**2-90**2 }{ 2 * 158.12 * 90 } ) = 28° 32'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 158.12**2-90**2-90**2 }{ 2 * 90 * 90 } ) = 122° 54'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 90**2-90**2-158.12**2 }{ 2 * 158.12 * 90 } ) = 28° 32'38" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3400 }{ 169.06 } = 20.11 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 90 }{ 2 * sin 28° 32'38" } = 94.18 ; ;





#2 Acute isosceles triangle.

Sides: a = 90   b = 86.01100434027   c = 90

Area: T = 3400
Perimeter: p = 266.0110043403
Semiperimeter: s = 133.0055021701

Angle ∠ A = α = 61.45659953123° = 61°27'22″ = 1.07326094633 rad
Angle ∠ B = β = 57.08880093755° = 57°5'17″ = 0.9966373727 rad
Angle ∠ C = γ = 61.45659953123° = 61°27'22″ = 1.07326094633 rad

Height: ha = 75.55655555556
Height: hb = 79.0610534456
Height: hc = 75.55655555556

Median: ma = 75.65662210467
Median: mb = 79.0610534456
Median: mc = 75.65662210467

Inradius: r = 25.56329445904
Circumradius: R = 51.22765699678

Vertex coordinates: A[90; 0] B[0; 0] C[48.90215135215; 75.55655555556]
Centroid: CG[46.30105045072; 25.18551851852]
Coordinates of the circumscribed circle: U[45; 24.4787775035]
Coordinates of the inscribed circle: I[46.99549782987; 25.56329445904]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.5444004688° = 118°32'38″ = 1.07326094633 rad
∠ B' = β' = 122.9121990625° = 122°54'43″ = 0.9966373727 rad
∠ C' = γ' = 118.5444004688° = 118°32'38″ = 1.07326094633 rad

Calculate another triangle

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 = 90 ; ; b = 86.01 ; ; c = 90 ; ;

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

p = a+b+c = 90+86.01+90 = 266.01 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 266.01 }{ 2 } = 133.01 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 133.01 * (133.01-90)(133.01-86.01)(133.01-90) } ; ; T = sqrt{ 11560000 } = 3400 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3400 }{ 90 } = 75.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3400 }{ 86.01 } = 79.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3400 }{ 90 } = 75.56 ; ;

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{ 90**2-86.01**2-90**2 }{ 2 * 86.01 * 90 } ) = 61° 27'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 86.01**2-90**2-90**2 }{ 2 * 90 * 90 } ) = 57° 5'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 90**2-90**2-86.01**2 }{ 2 * 86.01 * 90 } ) = 61° 27'22" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3400 }{ 133.01 } = 25.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 90 }{ 2 * sin 61° 27'22" } = 51.23 ; ;




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