4.2 Circles, Coordinate Geometry, and Solids

Circles, Coordinates, and 3-D Measurement

The ACT geometry description includes circles, surface area, volume, composite objects, and conic sections. That means the test can move from a simple radius question to a coordinate-plane circle or a solid built from familiar parts. The underlying skills are still compact: know the formulas, identify the given measurement, and keep units straight.

Circle questions are especially vulnerable to radius-diameter mistakes. The radius goes from the center to the circle. The diameter goes across the circle through the center, so diameter = 2r. Circumference is 2 pi r or pi d. Area is pi r squared. If the question gives diameter 14 and asks for area, the radius is 7 and the area is 49 pi, not 196 pi.

Arcs and sectors are fractional pieces of a circle. The fraction is central angle/360 degrees. Arc length equals that fraction times circumference. Sector area equals that fraction times circle area. For example, a 90-degree sector is one-fourth of the circle. If the radius is 8, the sector area is one-fourth of 64 pi, or 16 pi.

Circle Formula Map

Coordinate Geometry

Coordinate geometry turns pictures into algebra. The slope between two points is change in y over change in x. Parallel nonvertical lines have the same slope. Perpendicular nonvertical lines have slopes that multiply to -1, so their slopes are negative reciprocals. The midpoint is found by averaging x-values and averaging y-values. The distance formula is just the Pythagorean theorem on the horizontal and vertical changes.

For points (x1, y1) and (x2, y2), distance is sqrt((x2 - x1)^2 + (y2 - y1)^2). Keep the subtraction order consistent, but do not worry which point comes first because squaring removes the sign. Midpoint is ((x1 + x2)/2, (y1 + y2)/2). If an answer choice has one coordinate averaged and the other copied, it is a setup error.

The standard circle equation is a coordinate-geometry favorite: (x - h)^2 + (y - k)^2 = r^2. The center is (h, k), not (-h, -k). The signs inside the parentheses look reversed because the expression measures distance from the center. A circle with center (3, -4) and radius 6 has equation (x - 3)^2 + (y + 4)^2 = 36. The radius is 6, but the right side is 36 because the formula stores r squared.

Solids and Composite Figures

Three-dimensional questions are usually built from prisms, cylinders, pyramids, cones, spheres, or composite solids. You do not need many formulas if you remember the structure. A prism or cylinder has volume = base area times height. A pyramid or cone has volume = one-third base area times height. A rectangular prism has volume length times width times height. A cylinder has volume pi r squared h because the base is a circle.

Surface area is different from volume. It counts exposed faces, so it uses square units. For a rectangular prism, add the areas of all six faces or use 2lw + 2lh + 2wh. For a cylinder, total surface area combines two circular bases with the curved rectangle wrapped around the side: 2 pi r squared + 2 pi rh. ACT questions may ask for lateral surface area, which excludes bases. Read that word carefully before adding circles.

Composite figures reward decomposition. If a solid is a cylinder topped by a cone, find the volume of each part and add them. If a rectangular shape has a smaller rectangle removed, subtract the missing area. When two pieces touch, the shared face may not count as exposed surface area. Draw a quick boundary around the visible outside before adding face areas.

Similarity works in three dimensions too. If all lengths in a solid are multiplied by k, surface area is multiplied by k squared and volume is multiplied by k cubed. A cube with side length doubled has surface area four times as large and volume eight times as large. ACT distractors often use the length scale for every measurement, so the unit type can eliminate wrong answers before calculation.

Worked Example

A cylinder has radius 3 and height 10. Its volume is base area times height: pi(3 squared)(10) = 90 pi. If the radius changes to 6 while height stays 10, the volume becomes pi(6 squared)(10) = 360 pi. Doubling only the radius quadruples the circular base area, so volume also quadruples.

Calculator and Scratchwork Strategy

ACT allows permitted calculators on the Math section, but official guidance also notes that all Math problems can be solved without one. Geometry is a good place to separate setup from computation. Write the formula first, substitute values second, and use the calculator only if arithmetic would slow you down. This prevents common calculator-friendly mistakes such as entering diameter where radius belongs or taking the square root of the right side of a circle equation too early.

Common Traps

• Using diameter in pi r squared without dividing by 2.
• Reading (x + 5)^2 as center x = 5 instead of x = -5.
• Reporting r squared as the radius in a circle equation.
• Mixing linear, square, and cubic units.
• Adding hidden or shared faces when a surface area question asks for only exposed area.
• Assuming a coordinate graph is drawn to scale instead of using slope, distance, or midpoint.