Geometry and Measurement: Scale Drawings, Circles, Area, Surface Area, Volume, and Angles
What this domain tests
Geometry and Measurement tests scale drawings, circles, area of composite figures, surface area, volume, and angle relationships. You need to choose the right formula, keep units consistent, and recognize when a scale factor must be squared or cubed.
Core rules
- Scale drawings: actual length = drawing length × scale factor. If the scale is 1 inch : 4 feet, a 3-inch segment represents 12 feet. Areas scale by the square of the linear scale factor.
- Circle: circumference C = 2πr = πd; area A = πr^2. The radius is half the diameter. Use 3.14 for π only when told, and label answers in square or linear units.
- Composite area: split the figure into rectangles, triangles, and semicircles; add the parts, or subtract a cutout from a larger shape.
- Surface area: find the area of every face using a net, then add. Rectangular prism: SA = 2lw + 2lh + 2wh.
- Volume of a prism or cylinder: V = Bh, where B is the area of the base. For a cylinder, B = πr^2.
- Angle pairs: complementary angles sum to 90°, supplementary angles sum to 180°, and vertical angles are equal. Adjacent angles share a side and vertex.
Worked example
A rectangular room measures 3 inches by 2 inches on a drawing with scale 1 inch : 4 feet. Find the actual area. Actual length: 3 × 4 = 12 feet. Actual width: 2 × 4 = 8 feet. Actual area: 12 × 8 = 96 square feet. A shortcut is to multiply the drawing area by 4^2 = 16: 6 × 16 = 96. Using 4 instead of 16 gives 24 square feet, a common error.
Traps
- Using the diameter as the radius in A = πr^2.
- Forgetting that area scale factors are squared and volume scale factors are cubed.
- Mixing inches and feet without converting.
- Confusing surface area (square units) with volume (cubic units).
- Treating vertical angles as supplementary when they are equal.