Calculations at a convex, axial-symmetric pentagon. This can be seen as a shape made of an isosceles trapezoid and an isosceles triangle, put together at the bases. There are two different shapes, one that widens towards the top and then tapers, and a second that narrows in two steps. The calculation is the same for both shapes. Between these two shapes is the house-shaped pentagon, where both sides with the length b are parallel to each other and perpendicular to the base a.
Enter the three side lengths and the single angle α and choose the number of decimal places. Then click Calculate. Please enter angles in degrees, here you can convert angle units.
Formulas:
d = √ 2c² - 2c² * cos( α )
e = √ b² - ( d/2 - a/2 )² + √ c² - ( d/2 )²
β = acos{ [ b² + c² - e² -(a/2)² ] / ( 2bc ) }
γ = ( 540° - α - 2β ) / 2
p = a + 2b + 2c
A = 1/4 * √ ( d + a )² * ( d - a + 2b ) * ( a - d + 2b ) + 1/2 * √( 4 * c² - d² ) / 4 * d
Lengths, width, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The axially symmetric pentagon is one type of a semi-regular pentagon. Unlike the regular pentagon, not all sides and angles are equal, only some. This type of pentagon is only symmetrical to its axis through the tip and through the middle of the base; this shape has no other symmetries. The calculation is carried out via the isosceles trapezoid and the isosceles triangle, which make up the shape. For the house shape, the lower part is not an isosceles trapezoid, but a rectangle, which of course simplifies the formulas enormously. The lower of the two shapes of the axially symmetrical pentagon, the one that tapers twice towards the top, can also be called a tent shape.
The less regular a polygon becomes and the more sides and corners it has, the more difficult it becomes to calculate and the more complicated the underlying formulas become. Even a relatively simple figure like this is very difficult to calculate.