Circles, Polygons, and Area

• Apply circumference = 2πr and area = πr² to circle problems
• Calculate arc length and sector area using the fraction (central angle)/360
• Apply area formulas for rectangles, parallelograms (base × height), and trapezoids (1/2 × (b₁ + b₂) × height)
• Compute the sum of interior angles of any polygon using (n − 2) × 180
• Apply the volume formulas for rectangular prisms, cylinders, cones, spheres, and pyramids

The ACT does not give you a formula sheet, so these must be memorized cold. Circles: circumference = 2πr (or πd), area = πr². An arc is a piece of the circumference, and a sector is a pizza-slice piece of the area. Both are scaled by the fraction (central angle)/360. So arc length = (θ/360) × 2πr and sector area = (θ/360) × πr². Quadrilateral areas: rectangle = length × width; parallelogram = base × height, where the height is the perpendicular distance between the bases, not the slanted side; trapezoid = 1/2 × (b₁ + b₂) × height, the average of the two parallel bases times the height. Polygons: the sum of interior angles is (n − 2) × 180, where n is the number of sides. A triangle (n=3) gives 180°, a quadrilateral (n=4) gives 360°, a pentagon (n=5) gives 540°, a hexagon (n=6) gives 720°. For a regular polygon (all sides equal and all angles equal), each interior angle is that sum divided by n. A regular hexagon inscribed in a circle splits into 6 equilateral triangles, so each side equals the radius: a circle with diameter 14 holds a hexagon with perimeter 6 × 7 = 42. For a shaded region, subtract: the whole shape's area minus each unshaded piece (a big circle minus a small circle minus a triangle). 3D volumes: rectangular prism = length × width × height. Cylinder = πr²h (area of the circular base times height). Cone = (1/3)πr²h. The cone is always one-third the cylinder with the same base and height. Sphere = (4/3)πr³. Pyramid = (1/3) × base area × height. Same one-third rule as the cone. Surface area is different: cylinder = 2πr² + 2πrh (two circles plus the wrap-around rectangle). Sphere surface area = 4πr². The ACT will sometimes give you a composite shape (a cylinder with a cone on top, or a box with a hemisphere carved out) and ask for combined volume. Break the composite into pieces, compute each, and add or subtract. Circle theorems for hard questions: an inscribed angle is half the central angle (and half the arc) it intercepts, so any angle inscribed in a semicircle is 90°; a tangent line is perpendicular to the radius at the point of tangency, which makes a right triangle with the radius; two tangent segments drawn from the same outside point are equal; and the perpendicular from the center to a chord bisects the chord. With the central angle θ in radians, arc length is rθ and sector area is (1/2)r²θ.

• A sector is just a fraction of a circle. Write (angle/360) × (area or circumference) and plug in.
• Cone and pyramid both have the one-third factor. If you forget it, you will be off by exactly 3x.
• For a regular hexagon, you can split it into 6 equilateral triangles. Often faster than the (n−2)·180 formula.
• In a parallelogram or trapezoid, the height is perpendicular to the base. A slanted side is never the height.
• Read composite-shape problems twice. Missing a 'with the hemisphere removed' clause is how students lose this question.