Triangle calculator

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

You have entered side a, b and area T.

Triangle has two solutions: a=5.7; b=5.7; c=11.31219125206 and a=5.7; b=5.7; c=1.41444380953.

#1 Obtuse isosceles triangle.

Sides: a = 5.7   b = 5.7   c = 11.31219125206

Area: T = 4
Perimeter: p = 22.71219125206
Semiperimeter: s = 11.35659562603

Angle ∠ A = α = 7.12772557231° = 7°7'38″ = 0.1244394079 rad
Angle ∠ B = β = 7.12772557231° = 7°7'38″ = 0.1244394079 rad
Angle ∠ C = γ = 165.7455488554° = 165°44'44″ = 2.89328044956 rad

Height: ha = 1.40435087719
Height: hb = 1.40435087719
Height: hc = 0.70772190477

Median: ma = 8.49113003973
Median: mb = 8.49113003973
Median: mc = 0.70772190477

Inradius: r = 0.35222380598
Circumradius: R = 22.97702523622

Vertex coordinates: A[11.31219125206; 0] B[0; 0] C[5.65659562603; 0.70772190477]
Centroid: CG[5.65659562603; 0.23657396826]
Coordinates of the circumscribed circle: U[5.65659562603; -22.26330333146]
Coordinates of the inscribed circle: I[5.65659562603; 0.35222380598]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 172.8732744277° = 172°52'22″ = 0.1244394079 rad
∠ B' = β' = 172.8732744277° = 172°52'22″ = 0.1244394079 rad
∠ C' = γ' = 14.25545114462° = 14°15'16″ = 2.89328044956 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 = 5.7 ; ; b = 5.7 ; ; c = 11.31 ; ;

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

p = a+b+c = 5.7+5.7+11.31 = 22.71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.71 }{ 2 } = 11.36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.36 * (11.36-5.7)(11.36-5.7)(11.36-11.31) } ; ; T = sqrt{ 16 } = 4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4 }{ 5.7 } = 1.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4 }{ 5.7 } = 1.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4 }{ 11.31 } = 0.71 ; ;

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{ 5.7**2-5.7**2-11.31**2 }{ 2 * 5.7 * 11.31 } ) = 7° 7'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.7**2-5.7**2-11.31**2 }{ 2 * 5.7 * 11.31 } ) = 7° 7'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.31**2-5.7**2-5.7**2 }{ 2 * 5.7 * 5.7 } ) = 165° 44'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4 }{ 11.36 } = 0.35 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.7 }{ 2 * sin 7° 7'38" } = 22.97 ; ;





#2 Acute isosceles triangle.

Sides: a = 5.7   b = 5.7   c = 1.41444380953

Area: T = 4
Perimeter: p = 12.81444380953
Semiperimeter: s = 6.40772190477

Angle ∠ A = α = 82.87327442769° = 82°52'22″ = 1.44664022478 rad
Angle ∠ B = β = 82.87327442769° = 82°52'22″ = 1.44664022478 rad
Angle ∠ C = γ = 14.25545114462° = 14°15'16″ = 0.2498788158 rad

Height: ha = 1.40435087719
Height: hb = 1.40435087719
Height: hc = 5.65659562603

Median: ma = 3.02204002322
Median: mb = 3.02204002322
Median: mc = 5.65659562603

Inradius: r = 0.62442958092
Circumradius: R = 2.87221933573

Vertex coordinates: A[1.41444380953; 0] B[0; 0] C[0.70772190477; 5.65659562603]
Centroid: CG[0.70772190477; 1.88553187534]
Coordinates of the circumscribed circle: U[0.70772190477; 2.7843762903]
Coordinates of the inscribed circle: I[0.70772190477; 0.62442958092]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 97.12772557231° = 97°7'38″ = 1.44664022478 rad
∠ B' = β' = 97.12772557231° = 97°7'38″ = 1.44664022478 rad
∠ C' = γ' = 165.7455488554° = 165°44'44″ = 0.2498788158 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 = 5.7 ; ; b = 5.7 ; ; c = 1.41 ; ;

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

p = a+b+c = 5.7+5.7+1.41 = 12.81 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.81 }{ 2 } = 6.41 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.41 * (6.41-5.7)(6.41-5.7)(6.41-1.41) } ; ; T = sqrt{ 16 } = 4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4 }{ 5.7 } = 1.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4 }{ 5.7 } = 1.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4 }{ 1.41 } = 5.66 ; ;

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{ 5.7**2-5.7**2-1.41**2 }{ 2 * 5.7 * 1.41 } ) = 82° 52'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.7**2-5.7**2-1.41**2 }{ 2 * 5.7 * 1.41 } ) = 82° 52'22" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.41**2-5.7**2-5.7**2 }{ 2 * 5.7 * 5.7 } ) = 14° 15'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4 }{ 6.41 } = 0.62 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.7 }{ 2 * sin 82° 52'22" } = 2.87 ; ;




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