Geometry

Paper Triangles in a Circle Maximising Area

Three equal right-angled triangles are inscribed in a circle. They are positioned around an equilateral triangle which stabilises the construction. The triangles are cut from paper, with dimensions noted. The right-angled triangles are 7″ x 9” = 17.8 cm x 22.3 cm, and the equilateral triangle has side length 2.46″ = 6.24 cm. The six outermost points are equidistant from the centre of the equilateral triangle, thus lying on a circle, which is not shown. Iterative modelling with Geogebra found that this construction maximises the area of the circle covered by the three right-angled triangles. The ratio 7:9 of side lengths is a rational approximation to a numerical value found experimentally.

February 17, 2022 by Geometry 2

Paper Triangles in a Circle

Three equal right-isosceles triangles are inscribed in a circle. They are positioned around an equilateral triangle which stabilises the construction. The triangles are cut from A4 paper, with dimensions noted. The right triangles have two sides of length 8.3″ = 210 mm, and the equilateral triangle has side length 7.44″ = 188 mm. The six outermost points are equidistant from the centre of the equilateral triangle, thus lying on a circle, which is not shown.

February 17, 2022 by Geometry 2

Golden Rectangles in a Right Triangle

This construction was created by experimenting with the Geogebra interactive geometry application. I began with the right-triangle with side lengths 3, 4, and 5. Then I constructed its incircle, which has unit radius. In each of the three corners I inscribed a smaller circle, tangent to the incircle. Considering the circle inscribed in the 3-5 corner, I observed that the numerical value of its radius is 1/φ2, where φ (phi) is the golden ratio, equal to (1+√5)/2. This enables construction of three golden rectangles as shown. The dimensions of the rectangles are 1/φ2 x 1/φ, 1/φ x 1, and 1 x φ.

February 17, 2022 by Geogebra Geometry 2

An Asymmetric Rolling Shape

This asymmetric rolling shape is based on the 3-4-5 right-triangle. The perimeter is constructed from three arcs of a semicircle, with total length π * r, where r is the common radius of curvature. The shape does not have constant width like the Reuleaux polygons. The construction was created using the Geogebra interactive geometry application. Using a semicircle of fixed radius, the chord lengths in ratio 3:4:5 were found by iterative adjustment.

February 15, 2022 by Geogebra Geometry 0