Triangle calculator - result

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

You have entered side a, c and angle β.

Acute scalene triangle.

Sides: a = 9   b = 7.32994460491   c = 10

Area: T = 31.82198051534
Perimeter: p = 26.32994460491
Semiperimeter: s = 13.16547230245

Angle ∠ A = α = 60.25985814894° = 60°15'31″ = 1.05217106496 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 74.74114185106° = 74°44'29″ = 1.30444838406 rad

Height: ha = 7.07110678119
Height: hb = 8.6832731257
Height: hc = 6.36439610307

Median: ma = 7.52439876192
Median: mb = 8.77989410041
Median: mc = 6.50884859755

Inradius: r = 2.41770508634
Circumradius: R = 5.18327010036

Vertex coordinates: A[10; 0] B[0; 0] C[6.36439610307; 6.36439610307]
Centroid: CG[5.45546536769; 2.12113203436]
Coordinates of the circumscribed circle: U[5; 1.36439610307]
Coordinates of the inscribed circle: I[5.83552769755; 2.41770508634]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 119.7411418511° = 119°44'29″ = 1.05217106496 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 105.2598581489° = 105°15'31″ = 1.30444838406 rad

Calculate another triangle




How did we calculate this triangle?

1. Input data entered: side a, c and angle β.

a = 9 ; ; c = 10 ; ; beta = 45° ; ;

2. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 9**2+10**2 - 2 * 9 * 10 * cos 45° } ; ; b = 7.33 ; ;
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 = 9 ; ; b = 7.33 ; ; c = 10 ; ;

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

p = a+b+c = 9+7.33+10 = 26.33 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.33 }{ 2 } = 13.16 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.16 * (13.16-9)(13.16-7.33)(13.16-10) } ; ; T = sqrt{ 1012.5 } = 31.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.82 }{ 9 } = 7.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.82 }{ 7.33 } = 8.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.82 }{ 10 } = 6.36 ; ;

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{ 7.33**2+10**2-9**2 }{ 2 * 7.33 * 10 } ) = 60° 15'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+10**2-7.33**2 }{ 2 * 9 * 10 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 60° 15'31" - 45° = 74° 44'29" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.82 }{ 13.16 } = 2.42 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9 }{ 2 * sin 60° 15'31" } = 5.18 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.33**2+2 * 10**2 - 9**2 } }{ 2 } = 7.524 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10**2+2 * 9**2 - 7.33**2 } }{ 2 } = 8.779 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.33**2+2 * 9**2 - 10**2 } }{ 2 } = 6.508 ; ;
Calculate another triangle

Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.