Question Video: Using the Sine Rule to Find the Area of a Triangle Inscribed in a Circle | Nagwa Question Video: Using the Sine Rule to Find the Area of a Triangle Inscribed in a Circle | Nagwa

# Question Video: Using the Sine Rule to Find the Area of a Triangle Inscribed in a Circle Mathematics • Second Year of Secondary School

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π is the center of a circle, and π΄, π΅, and πΆ are points on the circumference. If π΅πΆ = 13 cm and πβ πΆππ΅ = 84Β°, find the area of the circle π, giving the answer to the nearest square centimeter.

02:30

### Video Transcript

π is the center of a circle, and π΄, π΅, and πΆ are points on the circumference. If π΅πΆ equals 13 centimeters and the measure of angle πΆππ΅ is 84 degrees, find the area of the circle π, giving the answer to the nearest square centimeter.

We know that the area of a circle is ππ squared. So really, this problem is about finding the radius of this circle. Letβs begin by putting the information weβve been given on the diagram. π΅πΆ is 13 centimeters, and the measure of angle πΆππ΅ is 84 degrees. We donβt know the lengths of ππΆ or ππ΅, but theyβre each the radius of the circle.

Now, there are numerous different approaches we could take. But one approach is to apply the law of cosines in the triangle πΆππ΅. This states that π squared equals π squared plus π squared minus two ππ cos π΄, where π and π represent two sides of a triangle and π΄ represents the included angle. In our triangle, π is 13 centimeters. The angle π΄ is 84 degrees. And the two sides which enclose this angle π΄ are each the radius of the circle π.

We can therefore form an equation. 13 squared equals π squared plus π squared minus two π squared cos of 84 degrees. We can solve this equation to find the value of π squared, which weβll then be able to substitute directly into our area formula. Factorizing the right-hand side of our equation by π squared, we have 169 equals π squared multiplied by two minus two cos of 84 degrees. Dividing through, we have that π squared is equal to 169 over two minus two cos 84 degrees. And weβll keep our value for π squared in this exact form.

We can then substitute this value of π squared into the area formula and evaluate on a calculator. Rounding our answer, and we have that the area of circle π to the nearest square centimeter is 296 square centimeters.

As I mentioned, there are in fact numerous approaches to this problem, which you can try out yourself if you wish. We couldβve applied the law of sines in triangle πΆππ΅. Or we couldβve divided it in half to form two right triangles and then used right-angle trigonometry.

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