Question

In the diagram above, each of the eight triangles is identical.The radius of the circle is 10 cm.a)Find the size of angle AÖB, marked e in the diagram.b)Find the total area of the shaded regions, giving the answer to 3 significantfigures.

Answer 1

Given a circle with centre o and inscribed 8 identical triangles with radius 10cm

(a) To Determine: The size of angle AOB

Solution:

Note that all the 8 identical tringles divide the circle into 8 equal parts with 8 equal angles.

Also note that sum of angle at the centre of the circle is 360 degrees

Given that one of the angle is angle AOB, then

$\begin{gathered} 8\times m\angle AOB=360^0(sum\text{ of angles at a point)} \\ m\angle AOB=\theta,So \\ 8\times\theta=360^0 \\ \theta=\frac{360^0}{8} \\ \theta=45^0 \end{gathered}$

Hence, angle AOB is equal to 45⁰

(b) To Determine: The total area of the shaded regions

The area of the shaded region is

$\text{area of shaded region=area of circle - area of the 8 identical triangles}$$\begin{gathered} A_{\text{circle}}=\pi r^2;r=10\operatorname{cm} \\ A_{\text{circle}}=\pi\times10^2 \\ A_{\text{circle}}=100\pi \\ A_{\text{circle}}=314.1592654\operatorname{cm}^2 \end{gathered}$$\begin{gathered} A_{\text{triangles}}=\frac{1}{2}r^2\sin \theta \\ A_{\text{triangles}}=\frac{1}{2}\times10^2\times\sin 45^0 \\ A_{\text{triangles}}=\frac{1}{2}\times100\times0.7071 \\ A_{\text{triangles}}=35.35533908\operatorname{cm}^2 \\ A_{8\text{triangles}}=8\times35.35533908\operatorname{cm} \\ A_{8\text{triangles}}=282.8427125\operatorname{cm}^2 \end{gathered}$$\begin{gathered} A_{\text{shaded region}}=A_{circle}-A_{8triangles} \\ A_{\text{shaded region}}=314.1592654\operatorname{cm}-282.8427125\operatorname{cm}^2 \\ A_{\text{shaded region}}=31.31655\operatorname{cm}^2 \\ A_{\text{shaded region}}=31.3\operatorname{cm}^2(3\text{significant figure)} \end{gathered}$

Answer Summary:

(a) The size of angle AOB is 45⁰

(b) The total area of the shaded region correct to 3 significant figures is 31.3cm²