Imagine taking a simple 2D curve, like a line or a parabola, and spinning it rapidly around an axis. The resulting 3D shape—like a cylinder, cone, or vase—is called a Surface of Revolution. Finding the surface area of these shapes is a classic problem in calculus. This calculator automates the heavy lifting, allowing you…
A Lemniscate is a plane curve with a characteristic “Figure-8” shape, often associated with the infinity symbol (∞). The name comes from the Latin word “lemniscatus,” meaning “decorated with ribbons.” While there are many variations, the two most famous types are the Lemniscate of Bernoulli (Polar) and the Lemniscate of Gerono (Algebraic). This calculator…
In many real-world scenarios, you do not have a perfect mathematical formula like y = x². Instead, you have a set of data points collected from an experiment, a survey, or a sensor. The Area of Graph Calculator is designed for this exact purpose: to calculate the area under a curve defined solely by discrete…
A Nonagon (also known as an Enneagon) is a polygon with 9 sides and 9 angles. While not as common as hexagons or octagons, nonagons appear in specialized architecture, coin designs, and geometric art. A Regular Nonagon has all sides equal in length and all interior angles equal to 140°. This calculator is designed…
The Koch Snowflake is one of the earliest and most famous fractal curves, described by Helge von Koch in 1904. It starts with an equilateral triangle, and in each iteration, the middle third of every line segment is replaced by a smaller equilateral triangle. This creates a paradoxical shape: it has a finite area…
In geometry, a ring is formally known as an Annulus (Latin for “little ring”). It is the region modeled by two concentric circles—a larger outer circle and a smaller inner hole. You see this shape everywhere: washers, donuts, gaskets, and even the planet Saturn! Calculating the area of a ring is essential in engineering…
A Torus is the mathematical name for the shape of a donut, a bagel, or an inner tube. It is a 3D surface generated by revolving a circle (the “tube”) around an axis coplanar with the circle. While “Donut” is the tasty common name, knowing the geometry of a Torus is crucial for everything…
When two circles intersect, they form a lens-shaped region in the middle. This region is mathematically known as a Vesica Piscis (when circles are equal) or an asymmetric lens (when radii differ). Calculating this area is a complex challenge in geometry because it requires finding the intersection points and summing the areas of two…
A Circular Segment is the “crust” of a pizza slice. It is the region bounded by a chord and the arc subtended by the chord. Specifically, a “Minor Segment” is the area less than a semicircle (Angle < 180°). If you cut a round loaf of bread with a straight knife, the smaller piece…
In geometry, a “washer” shape is formally called an Annulus. It is the region bounded by two concentric circles. It represents the flat face of a metal washer, a DVD, or a donut’s 2D cross-section. Calculating this area is critical in mechanical engineering for determining load distribution, material costs for stamped parts, and pressure analysis….