Answer: I would paint either of the 3/4 of a circle or a circular ring.
Step-by-step explanation: We shall start by calling the three quarter circle on the left A, and the ring on the right we shall call B.
The area of both can be calculated and that figure which results in the lesser value (of area) shall cost you less to paint.
For A, the area of the circle is given as
Area = πr²
Where π shall be taken as 3.14 and the radius r is 6.
Area = 3.14 * 6²
Area = 3.14 * 36
Area = 113.04
The area of the quarter of the circle would be subtracted from the total area as shown in the calculation above.
Area of sector = ∅/360 * πr²
The sector is a quarter of the circle and that makes the angle at the sector to be equal to (360/4) 90 degrees. Therefore ∅ equals 90.
Area of sector = (90/360) * 113.04
Area of sector = 1/4 * 113.04
Area of sector = 28.26
Area of A therefore is derived as;
Area of A = 113.04 - 28.26
Area of A = 84.78 square metres
For B, the area of the ring is given as follows;
Area of B = Area of larger circle - Area of smaller circle
Area of B = πR² - πr²
Where R is the radius of the larger circle and r is the radius of the smaller inner circle. By factorizing the right hand side of the equation, we now have;
Area of B = π (R² - r²)
Area of B = 3.14 (6² - 3²)
Area of B = 3.14 (36 - 9)
Area of B = 3.14 * 27
Area of B = 84.78 square metres.
From the calculations shown along with the explanations, both the three-quarters and the ring have exactly the same area, which means you are going to spend exactly the same quantity of paint and spend the same amount of time painting either one of them.
So the answer is; I would paint EITHER ONE OF THEM.
