The shape in the diagram shows a circle centered at 𝐶, from which a segment has been removed. Find the area of the shaded shape. Give your answer to two significant figures. Show all your working.
If we label the two points 𝐴, 𝐵, as shown in the diagram, we can see that the shaded shape is made up of two different shapes, a pink triangle 𝐴𝐵𝐶 and a yellow sector, also 𝐴𝐵𝐶. The shaded area is equal to the area of the triangle plus the area of the sector. The area of any triangle can be calculated using the formula: a half 𝑎𝑏 multiplied by sin 𝐶. We know that the lengths of the sides 𝐴𝐶 and 𝐵𝐶 are equal to 𝑟, as they are the radii of the circle. We will let the angle inside the triangle where these two radii or radiuses meet be 𝛼.
The lower case letters in the formula are the sides. And the capital 𝐶 is the angle. As 𝑟 multiplied by 𝑟 is equal to 𝑟 squared, the formula for the area of this triangle is a half 𝑟 squared multiplied by sin 𝛼. The formula for the area of a sector is 𝜃 divided by 360 multiplied by 𝜋𝑟 squared, where 𝜃 is the angle inside the sector. In this case, it is the reflex angle 𝐴𝐶𝐵.
In order to work out the area of these two shapes, we firstly need to calculate the unknowns, 𝛼, 𝑟, and 𝜃. The triangle 𝐴𝐵𝐶 is isosceles, since the length of 𝐴𝐶 is equal to the length of 𝐵𝐶. This means that the angle 𝐶𝐵𝐴 is equal to the angle 𝐶𝐴𝐵. We already know that angle 𝐶𝐵𝐴 is 35 degrees. Therefore, angle 𝐶𝐴𝐵 must also be 35 degrees.
We know that angles in a triangle sum or add up to 180 degrees. This means that 𝛼 plus 35 plus 35 must equal 180. 35 plus 35 is equal to 70. So 𝛼 plus 70 must equal 180. Subtracting 70 from both sides of this equation gives us 𝛼 equals 110, as 180 minus 70 equals 110. The angle 𝛼 inside the triangle 𝐴𝐵𝐶 is equal to 110 degrees.
We also know that angles in a circle, or at a point, sum to 360 degrees. This means that 𝜃 plus 𝛼 equals 360. As we know 𝛼 is equal to 110, 𝜃 plus 110 must equal 360. Subtracting 110 from both sides of this equation gives us a value of 𝜃 equal to 250 degrees. This is because 360 minus 110 equals 250. We have now worked out two of the unknowns, 𝛼 and 𝜃. We now need to work out the length of the radius of the circle.
We could split the triangle 𝐴𝐵𝐶 into two right-angled triangles. We could then use right-angled trigonometry or SOHCAHTOA to work out the length 𝑟. However, as we know that 12 centimeters is opposite the 110 degrees, we can you use the sine rule. The sine rule states that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵. The lowercase letters are the lengths and the capital letters are the angles. Substituting in the values from our triangle gives us 𝑟 over sin 35 is equal to 12 over sin 110.
To calculate 𝑟, we can multiply both sides by sin 35. This gives us 𝑟 is equal to 12 divided by sin of 110 multiplied by sin of 35. Typing this into our calculator gives us an answer for 𝑟 equal to 7.3246 and so on. For accuracy, it’s important at this stage that we don’t round our answer and that we use the full calculator display in the remainder of our calculations. We have now worked out all three of the unknowns. 𝛼 equals 110 degrees. 𝜃 equals 250 degrees. And 𝑟 equals 7.3246 and so on centimeters. We can now substitute these values into the two formulas.
The area of the triangle is equal to a half multiplied by 7.3246 squared multiplied by sine of 110 degrees. Typing this into the calculator gives us an area of 25.2074 and so on centimeters squared. The area of the sector is equal to 250 over 360 multiplied by 𝜋 multiplied by 7.3246 squared. Using the full answer on the calculator for the radius gives us the area of the sector equal to 117.0471 and so on. The shaded area can then be calculated by adding these two values: 25.2074 and 117.0471. This gives us a total of 142.2546 and so on centimeters squared.
We were asked to give our answer to two significant figures. The first significant figure is the one in the hundreds column. The second one is the four in the tens column. This means that the two in the units column is our deciding number. As this is less than five, we will round down. Rounding down gives us a final answer of 140 centimeters squared, to two significant figures.