Search Results for “circles”
Crop Circles
Farmers usually plant their crops in circular fields because the irrigation process is much more efficient that way. But most plots of land are rectangular, begging the question how can circles fill a rectangular plot of land most efficiently? This app lets you experiment with different configurations and sizes when using four crop circles to cover a square plot of land. The rules: all four circles must be entirely contained within the square, and the circles cannot overlap each other in any way (obviously). The total area of the four circles combined is calculated at the bottom so that you can easily determine which configuration is the best without manually calculating the areas of the circles. Concept inspired by ”The Farmer and the Mathematician: Using Geometry Expressions and Google Earth to Investigate Crop Circles”, by Larry Ottman. The book explores this subject in intriguing depth if you’re interested…
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!
Archimedes Twins
The large circle is tangent to each of the purple circles, which are tangent to each other. The radius of the large circle is the sum of the radii of the purple circles. The Yellow circles are each tangent to one of the purple circles, to the large circle and to the common tangent of the two purple circles. The yellow circles are called the Archimedes Twins, and have the same radius. Can you work out what this radius is in terms of the radii of the purple circles?
Circle Radius and Proportional Position Puzzler
Five circles are centered proportionally around an ellipse according to functions of the variable r. Each circle’s radius is defined by the same function as its position, of r. Can you figure out the five functions of r which define the five circles? Download the .gx source for the answer. Tip: if you press the ”go” button, r will animate at a constant rate and the ellipse will stay a constant size. If your drag the slider, note that the ellipse is not changing size or position, but the app is zooming in and out in order to display everything within its window most efficiently. Hint: one circle’s radius and position are defined by f(r) = r.
Hendecagon Diagonals, Circles and Tangents
Before even attempting to understand this app, take a look at the heptagon and nonagon versions. It’s the same situation here, circles centered at intersection points of diagonals within the hendecagon. Drag the green points to resize the circles. Resize the circles so that they are tangent to at least 4 diagonals at the same time (this case is possible in at least two positions for each circle). How many instances can you find on this one? Notice a trend with this and the other versions? Now that you've got this one, check out the final installment, the tridecagon version.
Circles, Tangents, and Heptagon Diagonals
Two circles are centered at intersection points of diagonals of a regular hepatgon. It turns out that circles centered at intersection points in regular polygons (particularly interestingly with polygons of odd numbers of sides) can be tangent to many other diagonals of that polygon. Try resizing the circles by dragging the green points. How many diagonals can each circle be tangent to? Ready for more? Check out the nonagon version!
Circle isotomic and Conchoid
The circle isotomic with respect to P is the locus of the reflection of P in the tangents of the circle. The Conchoid is the locus of the points a given fixed distance from a point on the circle, and lying on a line through that point and P.
Circle isotomic
The circle isotomic is the locus of the reflections of a given point in the tangents to the circle. It is easy to convince yourself that it is also the envelope of the circles whose centers lie on the circle and which pass through the given point. The given point is C. Try dragging it outside the circle.
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.
