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Angle Subtended by an Arc of a Circle on the Circle and at the Center

In the given ...

Question

In the given figure, AB is a side of a regular six-sided polygon and AC is a side of a regular eight-sided polygon inscribed in the circle with centre O. Calculate the sizes of :

(i) $∠$ AOB,

(ii) $∠$ ACB,

(iii) $∠$ ABC.

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Solution

(i) Since $AB$ is the side of a regular hexagon

$\angle AOB = 60^o$

(ii) $\widehat{AB}$ subtends $\angle ACB$ at the circumference and $\angle AOB$ at the center

We know $\angle ACB = \frac{1}{2} \angle AOB = \frac{60}{2} = 30^o$

(iii) Since $AC$ is the side of a regular octagon

$\angle AOC = \frac{360}{8} = 45^o$

$\widehat{AC}$ subtends $\angle ABC$ at the circumference and $\angle AOC$ at the center

Therefore $\angle ABC = \frac{1}{2} \angle AOC = \frac{45}{2} = 22.5^o$

$\\$

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