Nearly every ACT circle question is built from two formulas: area equals pi times the radius squared, and circumference equals 2 pi times the radius. Arcs, sectors, and shaded regions are those two formulas with a fraction attached. Get comfortable moving between the radius, the diameter, the area, and the circumference, and this topic turns into fast points.
The two formulas everything else grows from
Written compactly, the area is A = pi r^2 and the circumference is C = 2 pi r. Circumference is also pi times the diameter, which is handy when a problem hands you the diameter directly. For a circle with a radius of 5, the area is 25 pi and the circumference is 10 pi. The ACT does not supply a formula sheet, so both of these need to live in your head.
Radius versus diameter, the mistake to kill first
More circle points are lost here than anywhere else. If a problem says a circle has a diameter of 14, the radius is 7, so the area is 49 pi. Using 14 as the radius gives 196 pi, which will be one of your four answer choices, waiting patiently. Build the habit of writing r = and its value on the figure before you touch a formula. It takes two seconds and removes a whole category of error.
Working backwards to the radius
Plenty of questions give you the area or the circumference and want something else. The move is always the same: solve for the radius first, then go where you need to go. If a circle has an area of 36 pi, then pi times the radius squared equals 36 pi, so the radius squared is 36 and the radius is 6, making the circumference 12 pi. Going the other direction, a circumference of 20 pi means 2 times the radius is 20, so the radius is 10 and the area is 100 pi.
Arcs and sectors are fractions of the whole
A central angle has its vertex at the center of the circle. The piece of the circumference it cuts off is an arc, and the wedge of the interior it cuts off is a sector. In both cases you take the whole thing and multiply by the central angle over 360.
- Arc length equals the central angle over 360, times 2 pi times the radius. For a radius of 12 and a central angle of 45 degrees, the full circumference is 24 pi and 45 over 360 is one eighth, so the arc length is 3 pi.
- Sector area equals the central angle over 360, times pi times the radius squared. For a radius of 6 and a central angle of 120 degrees, the full area is 36 pi and 120 over 360 is one third, so the sector area is 12 pi.
Say the fraction out loud as you set it up. A 90 degree sector is a quarter of the circle, and a 60 degree sector is a sixth. If your sector comes out bigger than the whole circle, you skipped the fraction.
The equation of a circle on the coordinate plane
A circle centered at the point (h, k) with radius r has the equation (x - h)^2 + (y - k)^2 = r^2. Two details trip students up: the signs inside the parentheses are subtracted, so the center coordinates come out with the opposite sign of what you see, and the number on the right is the radius squared, not the radius.
Given (x - 3)^2 + (y + 2)^2 = 49, the center is at (3, negative 2) and the radius is 7, since 49 is 7 squared. The y + 2 is really y minus negative 2. For more on how points and lines behave on the grid, see coordinate geometry on the ACT.
Circles combined with other shapes
Harder questions bury a circle inside another figure and ask for a shaded region. The strategy is subtraction. If a circle is inscribed in a square with a side of 10, the circle touches all four sides, so its diameter is 10 and its radius is 5. The square's area is 100, the circle's area is 25 pi, and the shaded corners come to 100 minus 25 pi, or roughly 21.5.
Notice that the answer stays in terms of pi unless the question asks for a decimal, so resist the urge to multiply it out early. Composite figures show up again in area and volume formulas on the ACT.
A quick self-check
Three to try. A circle has a diameter of 18; find its area. A circle with a radius of 10 has a sector with a central angle of 72 degrees; find the sector area. A circle has the equation (x + 4)^2 + (y - 1)^2 = 25; find the center and radius. The answers are 81 pi, since the radius is 9; 20 pi, since 72 over 360 is one fifth of 100 pi; and a center of (negative 4, 1) with a radius of 5.
If the sector question was the slow one, repetition is the fix, and a focused set of circle questions on thirty-six gets that step down to reflex. For where circles sit among the other geometry topics, see how much geometry is on the ACT.
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