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?

1. Input data entered: side a, c and area S.

a = 90 ; ; c = 90 ; ; S = 3400 ; ;

2. From area T and side a we calculate height ha - The area of the triangle is the product of the length of the base and the height divided by two:

T = fraction{ a h_a }{ 2 } ; ; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 3400 }{ 90 } = 75.556 ; ;


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 ; ;

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

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

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 158.12**2+90**2-90**2 }{ 2 * 158.12 * 90 } ) = 28° 32'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90**2+90**2-158.12**2 }{ 2 * 90 * 90 } ) = 122° 54'43" ; ; gamma = 180° - alpha - beta = 180° - 28° 32'38" - 122° 54'43" = 28° 32'38" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 158.12**2+2 * 90**2 - 90**2 } }{ 2 } = 120.524 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 90**2 - 158.12**2 } }{ 2 } = 43.005 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 158.12**2+2 * 90**2 - 90**2 } }{ 2 } = 120.524 ; ;







#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?

1. Input data entered: side a, c and area S.

a = 90 ; ; c = 90 ; ; S = 3400 ; ; : Nr. 1

2. From area T and side a we calculate height ha - The area of the triangle is the product of the length of the base and the height divided by two:

T = fraction{ a h_a }{ 2 } ; ; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 3400 }{ 90 } = 75.556 ; ; : Nr. 1


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 ; ;

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

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

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 86.01**2+90**2-90**2 }{ 2 * 86.01 * 90 } ) = 61° 27'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90**2+90**2-86.01**2 }{ 2 * 90 * 90 } ) = 57° 5'17" ; ; gamma = 180° - alpha - beta = 180° - 61° 27'22" - 57° 5'17" = 61° 27'22" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 86.01**2+2 * 90**2 - 90**2 } }{ 2 } = 75.656 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 90**2 - 86.01**2 } }{ 2 } = 79.061 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 86.01**2+2 * 90**2 - 90**2 } }{ 2 } = 75.656 ; ;
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