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?

1. Input data entered: side a, b and area T.

a = 5.7 ; ; b = 5.7 ; ; T = 4 ; ;

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 * 4 }{ 5.7 } = 1.404 ; ;
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 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 5.7**2+11.31**2-5.7**2 }{ 2 * 5.7 * 11.31 } ) = 7° 7'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.7**2+11.31**2-5.7**2 }{ 2 * 5.7 * 11.31 } ) = 7° 7'38" ; ;
 gamma = 180° - alpha - beta = 180° - 7° 7'38" - 7° 7'38" = 165° 44'44" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.7**2+2 * 11.31**2 - 5.7**2 } }{ 2 } = 8.491 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.31**2+2 * 5.7**2 - 5.7**2 } }{ 2 } = 8.491 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.7**2+2 * 5.7**2 - 11.31**2 } }{ 2 } = 0.707 ; ;



#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?

1. Input data entered: side a, b and area T.

a = 5.7 ; ; b = 5.7 ; ; T = 4 ; ; : 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 * 4 }{ 5.7 } = 1.404 ; ; : 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 = 5.7 ; ; b = 5.7 ; ; c = 1.41 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 5.7**2+1.41**2-5.7**2 }{ 2 * 5.7 * 1.41 } ) = 82° 52'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.7**2+1.41**2-5.7**2 }{ 2 * 5.7 * 1.41 } ) = 82° 52'22" ; ;
 gamma = 180° - alpha - beta = 180° - 82° 52'22" - 82° 52'22" = 14° 15'16" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.7**2+2 * 1.41**2 - 5.7**2 } }{ 2 } = 3.02 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 5.7**2 - 5.7**2 } }{ 2 } = 3.02 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.7**2+2 * 5.7**2 - 1.41**2 } }{ 2 } = 5.656 ; ;
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