Geometric Measurement and Dimension
Geometric Measurement and Dimension
This domain tests area, surface area, volume, cross-sections, and scaling for two- and three-dimensional figures. Regents items often involve composite solids, density, or comparing a solid to its model.
Core rules
- Prism or cylinder volume: V equals B times h, where B is the base area. Pyramid or cone volume: V equals one-third B times h.
- Sphere: V equals four-thirds pi r cubed and SA equals 4 pi r squared. Cone lateral area: LA equals pi r l, where l is slant height; total SA equals pi r squared plus pi r l.
- Cylinder lateral area: LA equals 2 pi r h; total SA equals 2 pi r squared plus 2 pi r h.
- Cavalieri's principle: solids with equal heights and equal cross-sectional areas at every level have equal volumes.
- A plane cross-section of a prism or cylinder parallel to the base is congruent to the base; other cuts can produce rectangles, triangles, trapezoids, or ellipses.
- When a solid is scaled by factor k, surface area scales by k squared and volume by k cubed.
Worked example
A right cone has radius 6 cm and height 8 cm. Find its volume and total surface area. The slant height is l equals the square root of 6 squared plus 8 squared, which is the square root of 100, or 10 cm. Volume equals one-third pi times 6 squared times 8, which is one-third pi times 36 times 8, or 96 pi cubic cm. Total surface area equals pi r squared plus pi r l, which is pi times 36 plus pi times 6 times 10, or 36 pi plus 60 pi, or 96 pi square cm. Notice that the numerical values match but the units differ.
Traps
Use slant height, not vertical height, in cone lateral area. Do not forget the one-third in pyramid and cone volumes. Diameter is not radius. Keep square units for area and cubic units for volume. When a solid is enlarged, do not multiply volume by k; multiply by k cubed. For composite solids, subtract overlapping or hollow parts rather than adding every face.