# Thread: trigonometry/ circle results help

1. ## trigonometry/ circle results help

hello,

A sector of a circle with radius 5cm and an angle of π/3 subtended at the centre is a cardboard. It is then curved around to form a cone. Find its exact surface area and volume.

2. Originally Posted by maay
hello,

A sector of a circle with radius 5cm and an angle of π/3 subtended at the centre is a cardboard. It is then curved around to form a cone. Find its exact surface area and volume.

The surface area of the cone is the same as the area of the sector.

Since the angle is $\displaystyle \frac{\pi}{3}$, and a full circle is $\displaystyle 2\pi$, this means that you have

$\displaystyle \frac{\frac{\pi}{3}}{2\pi} = \frac{1}{6}$ of the circle.

So $\displaystyle A = \frac{1}{6}\pi r^2$

$\displaystyle = \frac{1}{6} \pi(5\,\textrm{cm})^2$

$\displaystyle = \frac{25\pi}{6}\,\textrm{cm}^2$.

3. Hello maay
Originally Posted by maay
hello,

A sector of a circle with radius 5cm and an angle of π/3 subtended at the centre is a cardboard. It is then curved around to form a cone. Find its exact surface area and volume.

The formula for the curved surface area of a cone is:
$\displaystyle A = \pi r l$
where $\displaystyle r$ is the radius and $\displaystyle l$ the slant-height of the cone.

So if the radius of the original cardboard circle is $\displaystyle 5$ cm, can you see that this means that $\displaystyle l = 5$ when the cardboard is formed into a cone?

So you can now:

• Use the formula above to find $\displaystyle r$, given that $\displaystyle A = \frac{25\pi}{6}$

• Use Pythagoras' Theorem to find $\displaystyle h$, the vertical height of the cone.

• Use the formula $\displaystyle V = \tfrac13\pi r^2h$ to find the volume of the cone.

Can you complete it now?