Or search by topic
![]() ![]() ![]() |
This model holly leaf is made in sections and joined together. Like a real holly leaf, it will not lie flat. It has negative curvature. To make the holly leaf, a circle centre C of radius 5 cm and radii CA and CB with \angle ACB = 125 degrees are drawn. The tangents to the circle at A and B meet at the point P. Eight identical 3 sided shapes are made by cutting along PA and PB and around the arc AB to make a 3 sided shape with 2 straight edges and one edge along the minor arc of the circle (the circles are thrown away). Two identical 4-sided shapes are made by drawing a circle with radius 5 cm, a diameter B*D* and tangents B*P* and D*Q*equal in length to PB. These shapes have edges B*P*, P*Q*, Q*D* and the semicircular arc (inside the rectangle) from B* to D*. The sketch shows (on a smaller scale) how the ten pieces are joined together to make the "holly leaf". Find the length of the boundary of the yellow area around P which is bounded by six arcs centred at P, each of radius r cm. All points on the boundary of the yellow region are equidistant from the point P. If the surface at P were flat, the boundary of the region would be a circle and its length would be 2\pi r. In this case the length of the boundary is greater than 2\pi r and the surface of the "holly leaf" has negative curvature at P. Compare the perimeter and area of this "holly leaf" with the similar flat leaf for which \angle ACB = 135 degrees. See the problem "Holly" for the flat version of this problem. What happens to the holly leaves as the angle \angle ACBchanges? [For positive curvature the boundary is less than 2 \pi r in length.] |
Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.
A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?