A sector of a circle has radius $12\text{ cm}$ and angle $135^\circ$. (i) Calculate the arc length of this sector, as a multiple of $\pi$. (ii) The sector is then used to make a cone. (a) Calculate the base radius $r$. (b) Calculate the height of the cone $h$.
The diagram shows a plant pot formed by cutting a small cone out of a larger cone and then attaching a circular base. (i) Find $l$. (ii) Calculate the total surface area of the outside of the plant pot. Use $A = \pi r l$ for curved surface area.
Some cones are mathematically similar. The mass $M$ grams is proportional to the cube of the base radius $r$ cm. One cone has mass $1458$ g and base radius $4.5$ cm. (i) Find an expression for $M$ in terms of $r$. (ii) Two cones have radii in the ratio $2:3$. Write down the ratio of their masses.