Draw to scale the arch in question onto the graph paper and superimpose a Cartesian plane on it, which includes a x-axis and an y-axis. Make sure the arch is above the x-axis and that the y-axis passes through its apex, which is either the highest point of the arch if pointing upward or the lowest point if facing downwards. This point is also referred to as its rise.
Establish the shape of the arch you are dealing with. The most common shape is a catenary, similar to the shape made of a chain held at its end if left suspended from two different positions. An arch is also built in the shape of an arc segment of a circle.
Divide the area under the curve made by the arch and the x-axis into segments using a ruler and pencil, then join each point of the segment lines touching on the curve to the next point to form trapeziums. Calculate the area of each trapezium under the curve by adding the height of the two lines parallel to the y-axis and dividing the sum by two, then multiplying the quotient by its length on the x-axis. Add up all the individual areas of the trapeziums to calculate the area under the arch.
Complete the outline of the arch with a pencil to form a full circle and determine its center. To calculate the length of the arch, draw two lines from the center touching each tip of the arch, then measure the angle between them using a protractor. Divide the angle you get by 360, then multiply the quotient by pi (3.142) and the diameter, which is double the length of the two lines.
Determine the length of the y-axis from the origin of the plane to the apex of the arch. You can calculate either the curvature of the arch, which is a constant, or the length of the arch given the other using the Cesàro equation, which equates the curvature to the length of the y-axis from the origin to the rise divided by the sum of the squared length of the arch and the squared length of the y-axis segment. Alternate the subject of the formula, depending on what you are calculating and what you already have in hand.