21  Functions II: Calculating Triangles

21.1 Functions within functions

Python functions can not only process Python statements, but they can also call user-defined functions. The implementation of Heron’s formula will serve as an example.

21.1.1 Calculating a Triangle’s Area Using Heron’s Formula

General Triangle.

To calculate the area \(T\) of a triangle, we use Heron’s formula

\[ T = \sqrt{s(s-a)(s-b)(s-c)} \]

where \(s\) denotes the semiperimeter. The semiperimeter is calculated by adding the lengths of the triangle’s sides and dividing the sum by two.

\[ s = \frac{1}{2} \cdot (a + b + c) \]

Coding Task

Implement a function area() to calculate the area of a triangle from its sides. Use a subfunction semiperimeter() to simplify the logic.

def semiperimeter(...):
    # TODO: implement the formula to calculate the semiperimeter.
    pass
    
def area(...):
    # TODO: implement the formula to calculate the area.
    pass
    
# testcase
print(area(3, 4, 5))
# should print 6.0 to the command line

def semiperimeter(a: float, b: float, c: float) -> float:
    """Calculates the semiperimeter of a triangle.
    
    Args:
        a: Side a of the triangle.
        b: Side b of the triangle.
        c: Side c of the triangle.
        
    Returns:
        s: The semiperimeter of the triangle.
    """
    return (a + b + c) / 2

def area(a: float, b: float, c: float) -> float:
    """Calculates the area of a triangle.
    
    The triangle's area is calculated from its sides using Heron's
    formula.
    
    Args:
        a: Side a of the triangle.
        b: Side b of the triangle.
        c: Side c of the triangle.
        
    Returns:
        T: The Area of the triangle."""
    s = semiperimeter(a, b, c)
    T = (s*(s-a)*(s-b)*(s-c))**(1/2)
    return T

print(area(3, 4, 5))
# should print 6.0 to the command line

6.0

21.1.2 The Altitude of a Triangle

Altitude of a Triangle.

The altitude \(h\) of a triangle is calculated by the formula

\[ h_a = \frac{2}{a}\sqrt{s(s-a)(s-b)(s-c)} \]

where \(s\) denotes the semiperimeter. To calculate the altitude \(h_b\) and \(h_c\),switch the respective variables in the formula accordingly.

Coding Task

Implement a function altitude() to calculate the altitude of a triangle from its sides. Use as many subfunctions as possible.

def altitude(...):
    # TODO: implement the formula to calculate the altitude of a
    # triangle.
    pass

# test
print(altitude(3, 4, 5, 'a'))
# should print 4.0 to the command line

def altitude(a: float, b: float, c: float, base: str = 'c') -> float:
    """Calculates the altitude of a triangle.
    
    The altitude is calculated to the base c by default. The base can
    be overridden if needed.
    
    Args:
        a: Side a of the triangle.
        b: Side b of the triangle.
        c: Side c of the triangle.
        base: The base of the altitude. The default value is 'c'
        
    Returns:
        h: The altitude of the triangle."""
    T = area(a, b, c)
    
    if base == 'c':
        h = (2*T)/c
    elif base == 'a':
        h = (2*T)/a
    else:
        h = (2*T)/b
    
    return h

print(altitude(3, 4, 5, 'a'))

NoteDefault Values for Function Arguments

If you have to draw the altitude of a triangle, you most likely would draw it from the horizontal side as a base line. Therefore, the base line will frequently be side \(c\) of your triangle. To account for this probability, Python offers a default value option for function arguments, as demonstrated in the altitude() function above.

The syntax requires that arguments with a default value must be placed at the end of the argument list. The default value is assigned using the \(=\) sign. This default value can be overridden either implicitly as a positional argument by providing a full list of arguments (e.g., altitude(3, 4, 5, 'a')), or explicitly as a keyword argument by directly assigning a new value to the parameter (e.g., altitude(3, 4, 5, base='a')).

21.1.3 Circumcircle of a Triangle

The Circumcircle of a Triangle.

The radius \(r\) of the circumcircle of a triangle is calculated by the formula

\[ r = \frac{abc}{4T} \]

where \(T\) denotes the area of the triangle.

Coding Task

Implement a function circumcircle() to calculate the circumcircle radius of a triangle from its sides. Use as many subfunctions as possible.

def circumcircle(...):
    # TODO: implement the formula to calculate the circumcircle of a
    # triangle
    pass

# test
print(circumcircle(3, 4, 5))
# should print 2.5 to the command line

def circumcircle(a: float, b: float, c: float) -> float:
    """Calculates the circumcircle radius of a triangle.
    
    Args:
        a: Side a of the triangle.
        b: Side b of the triangle.
        c: Side c of the triangle.
        
    Returns:
        r: The radius of the circumcircle of a triangle.
    """
    T = area(a, b, c)
    n = a * b * c
    d = 4 * T
    r = n / d
    return r

print(circumcircle(3, 4, 5))

21.1.4 The Incircle of a Triangle

The Incircle of a Triangle.

The incircle radius \(\varrho\) (small greek letter rho) of a triangle is calculated by the formula

\[ \varrho = \frac{T}{s} \]

where \(T\) denotes the area of the triangle and \(s\) the semiperimeter respectively.

Coding Task

Implement a function incircle() to calculate the incircle radius of a triangle from its sides. Use as many subfunctions as possible.

def incircle(...):
    # TODO: implement the formula to calculate the incircle radius
    pass

#test
print(incircle(3, 4, 5))
# should print 1.0 to the command line

def incircle(a: float, b: float, c: float) -> float:
    """Calculates the incircle radius.
    
    Args:
        a: Side a of the triangle.
        b: Side b of the triangle.
        c: Side c of the triangle.
        
    Returns:
        rho: Incircle radius of the triangle.
    """
    T = area(a, b, c)
    s = semiperimeter(a, b, c)
    rho = T/s
    return rho

print(incircle(3, 4, 5))