Cones questions with answers

Questions

1. Question 1Volume of a cone

   The diagram shows a traffic cone of height $12$ cm and base diameter $7$ cm. Find its volume, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{7}{2}$$= 3.5$
   2. Step 2: Volume of a cone = ⅓πr²h$V = \dfrac{1}{3}\pi r^2 h$$V = \dfrac{1}{3} \times \pi \times 3.5^2 \times 12$$= 49\pi$$= 153.9380\ldots$$= 154$ cm³ (3 s.f.)
2. Question 2Curved surface area

   The diagram shows a solid cone with base radius $5$ cm and slant height $8$ cm. Find its curved surface area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Curved surface area of a cone = πrl (from the formula sheet)Curved surface area $= \pi r l$Curved surface area $= \pi \times 5 \times 8$$= 40\pi$$= 125.6637\ldots$$= 126$ cm² (3 s.f.)
3. Question 3Total surface area

   The diagram shows a cone with base radius $3$ cm and slant height $9$ cm. Find its total surface area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Curved surface area of a cone = πrl (from the formula sheet)Curved surface area $= \pi r l$Curved surface area $= \pi \times 3 \times 9$$= 27\pi$
   2. Step 2: Total surface area: add the circular base, πr²Total surface area $= \pi r l + \pi r^2$Base area $= \pi \times 3^2$Total surface area $= \pi \times 3 \times 9 + \pi \times 3^2$$= 36\pi$$= 113.0973\ldots$$= 113$ cm² (3 s.f.)
4. Question 4Slant height by Pythagoras, then surface area

   The diagram shows a solid cone with base radius $3$ cm and height $8$ cm. Find its slant height $l$ cm, and hence its curved surface area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Slant height: the radius, the height and the slant height make a right-angled triangle$l^2 = 3^2 + 8^2$$= 73$$l = \sqrt{73}$$= 8.5440\ldots$
   2. Step 2: Curved surface area of a cone = πrl (from the formula sheet)Curved surface area $= \pi r l$Curved surface area $= \pi \times 3 \times 8.5440\ldots$$= 80.5253\ldots$$= 80.5$ cm² (3 s.f.)
5. Question 5Height from the volume

   The diagram shows a conical funnel with base radius $9$ cm. Its volume is $270\pi$ cm³. Find its height, $h$ cm.

   Show the worked solution

   1. Step 1: Volume = ⅓πr²h: write an equation in h$\dfrac{1}{3} \times \pi \times 9^2 \times h = 270\pi$
   2. Step 2: Solve for h$h = \dfrac{3 \times 270}{81}$$= 10$ cm
6. Question 6A cone made from a sector

   The diagram shows a sector of a circle of radius $24$ cm and angle $216^\circ$ ($\dfrac{3}{5}$ of the circle). Its two straight edges are joined, without overlapping, to make the curved surface of a cone. Find the height of the cone.

   Show the worked solution

   1. Step 1: The arc of the sector becomes the circumference of the cone's base$\dfrac{216}{360} \times 2\pi \times 24 = 2\pi r$$r = 14.4$ cm
   2. Step 2: The radius of the sector becomes the slant height: h² = l² − r²$h^2 = 24^2 - 14.4^2$$= 368.64$$h = \sqrt{368.64}$$= 19.2$ cm
7. Question 7Equal volumes: h in terms of r

   A solid cylinder of radius $r$ cm and height $2r$ cm is melted down and recast into a solid cone of base radius $\dfrac{3}{2}r$ cm and height $h$ cm, with no metal lost. Express $h$ in terms of $r$.

   Show the worked solution

   1. Step 1: Volume of the cylinder: πr²h with height 2rVolume $= 2\pi r^3$
   2. Step 2: Volume of the cone of radius 3/2r and height hVolume $= \dfrac{1}{3}\pi \left(\dfrac{3}{2}r\right)^2 h$$= \dfrac{3}{4}\pi r^2 h$
   3. Step 3: The volumes are equal: divide both sides by πr² (they cancel)$\dfrac{3}{4}\pi r^2 h = 2\pi r^3$$h = \dfrac{8}{3}r$
8. Question 8Half a cone: height from the volume, total surface area

   The diagram shows a doorstop that is half of a right circular cone, cut along a plane through its axis. Its base radius is $3$ cm and its volume is $18\pi$ cm³. Find its height, $h$ cm.

   Show the worked solution

   1. Step 1: Half a cone: volume = ½ × ⅓πr²h. Write an equation in h$\dfrac{1}{2} \times \dfrac{1}{3} \times \pi \times 3^2 \times h = 18\pi$
   2. Step 2: The π cancels: solve for h$\dfrac{9}{6}h = 18$$= 12$ cm
9. Question 9Equal volumes, then the surface area in surd form

   A solid cone has base radius $r$ cm and height $h$ cm. Its volume is equal to the volume of a solid cylinder of radius $r$ cm and height $2r$ cm. Find $h$ in terms of $r$. Hence show that the total surface area of the cone is $k\pi r^2$ cm², and find $k$, leaving your answer in surd form.

   Show the worked solution

   1. Step 1: Equal volumes: ⅓πr²h = the other volume. Divide both sides by ⅓πr²$\dfrac{1}{3}\pi r^2 h = 2\pi r^3$$h = 6r$
   2. Step 2: Slant height: l² = r² + h²$l^2 = r^2 + (6r)^2$$\phantom{l^2} = 37r^2$$l = \sqrt{37}\,r$
   3. Step 3: Total surface area = πrl + πr²: take out πr²$\pi r \times \sqrt{37}\,r + \pi r^2 = (1 + \sqrt{37})\pi r^2$