Study guide: Solve Geometry with Areas, Volumes, Nets, and Coordinates
Use perpendicular heights, fractional prism lengths, surface-area nets, and coordinate polygons.
Ideas to explain in your own words
- A parallelogram can be rearranged into a rectangle with the same base and perpendicular height.
- The prism volume formula applies to fractional lengths as well as whole lengths.
- Surface area counts the flat areas of a solid’s boundary faces.
- A polygon can be specified by its ordered vertices on a coordinate plane.
Useful vocabulary
- perpendicular height
- The shortest distance from a base line to the opposite level.
- net
- A flat arrangement of faces that can fold into a solid.
Watch your reasoning
- A triangle needs half the corresponding base-by-height rectangle area.
- Volume has cubic units even when its value is a fraction.
- Surface area is square units; volume is cubic units.
- A coordinate is a signed position; a side length is nonnegative.
A worked review example
A triangle has base 10 cm and perpendicular height 7 cm. Its area is (10×7)/2=35 square cm. A corresponding parallelogram has area seventy square cm. A slanted edge length, even if shown as nine centimeters, is not the height used in this calculation. Check that the height is perpendicular to the base line. A triangle needs half the corresponding base-by-height rectangle area. Draw the base line and its right-angle height marker before computing area.
Make a practice plan
- Use Perpendicular Height for Triangle and Parallelogram Area: A triangle with base 13 and perpendicular height 4 has area 26 square units.
- Calculate Prism Volume with Fractional Edge Lengths: Edges 1/2,3/4,2 units give volume 3/4 cubic unit.
- Find Surface Area from Every Face in a Net: Edges 3,5,7 give surface area 142 square units and volume 105 cubic units.
- Find Axis-Aligned Polygon Sides from Coordinates: Rectangle vertices (−3,−1),(2,−1),(2,4),(−3,4) give side lengths 5 and 5, area 25.