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Search Results for “volume-around-an-axis”

By Phil Todd
Tchirnhausen's Cubic
The caustic formed by light projecting perpendicular to the axis of a parabola is called Tchirnhausen's Cubic. What happens when the light projects at some other angle?

Tags: Parabola, caustic

By Nick Halsey
Geometric Top View Model of the Twisted Savonius Wind Turbine (Interactive)
This app models the top view of the Twisted Savonius Vertical Axis Wind Turbine (VAWT). The various inputs and draggable points allow you to see how the model can trace the blades' surfaces. You can also control the twist angle, radius, and rotation - which makes the whole thing spin! Learn more about the Twisted Savonius Modeling Project here.

Tags: Twisted-Savonius, Top-View, Model, Geometric, Real-World, Circles, Arcs, Loci

By Nick Halsey
Basic Unit Circle
This very basic representation of the unit circle displays the unit circle with an input for the standard angle θ in degrees (which controls the angle between the hypotenuse and the x axis). The outputs represent the other two sides of the triangle and give their lengths through decimals. A good investigation for geometry students is to have them test out different angles here, then compare the results to those testing the angles with sine and cosine on their calculators. This allows them to visualize the unit circle in a precise diagram rather than simply running inputs and outputs on their calculators.

Tags: Geomtery, Unit-Circle, Sine, Cosine

By Nick Halsey
Circles, Tangents and Nonagon Diagonals
You may want to see the heptagon version before attempting this one Every diagonal within a regular nonagon is drawn. Circles are centered at each intersection of diagonals along a vertical axis (these same constructions can be made nine times around the nonagon). Each circle can be tangent to at least 4 diagonals when the circle is at least 2 different sizes. Unnecessary diagonals have been hidden. Drag the green points to resize the circles. Can you find all 13 positions where a circle is tangent to at least 4 diagonals? Hint: sometimes the circle is not entirely contained within the nonagon. Ready for more? Check out the hendecagon version!

Tags: Nonagons, Circles, Diagonals, Tangents, Puzzler

By Nick Halsey
Area Under Sine (draggable)
Observe the definite integral of sine, or the area between the function sin(x) and the x axis, and how it changes between different bounds by dragging the boundaries, a and b. What happens to the area when the interval is 2Π? Why?

Tags: Calculus, Defininte-Integral, Sine, Draggable

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