Parabolic Arc Calculator

What is a Parabolic Arc?

A parabolic arc is like a smile on a happy face! It's a curved line that's part of a parabola, which is a special U-shaped curve. You can see parabolic arcs in many places, like the path of a ball when you throw it or the shape of a satellite dish.

How to Calculate a Parabolic Arc

To understand a parabolic arc, we need to know two important things: how wide it is (we call this the chord length) and how tall it is (we call this the height). Once we know these, we can figure out cool things like how much space the arc covers (its area) and how long the curved line is (its arc length).

Formula

For a parabolic arc, we use these formulas:

Area: \[ A = \frac{2}{3}bh \]

Arc Length: \[ L \approx b + \frac{4h^2}{3b} \]

Where:

• \(A\) is the area of the parabolic segment
• \(L\) is the length of the parabolic arc
• \(b\) is the chord length (how wide the arc is)
• \(h\) is the height (how tall the arc is)

Calculation Steps

1. Measure the chord length (b) and height (h) of your parabolic arc
2. To find the area:
   • Multiply the chord length by the height
   • Multiply this result by 2/3
3. To find the arc length:
   • Square the height (multiply it by itself)
   • Multiply this by 4, then divide by 3 times the chord length
   • Add this result to the chord length

Example and Visual Representation

Let's imagine a parabolic arc with a chord length of 6 units and a height of 2 units:

Now, let's calculate:

Area: \(A = \frac{2}{3} \times 6 \times 2 = 8\) square units

Arc Length: \(L \approx 6 + \frac{4 \times 2^2}{3 \times 6} \approx 6.89\) units

This parabolic arc covers an area of about 8 square units and has a length of about 6.89 units along its curve.