Math · Geometry & data
Geometry: Perimeter, Area, and Volume
Estimated time includes reading and one quiz. Take the time you need.
Start here: key ideas
- Perimeter measures a boundary; area covers a region; volume fills a solid.
- Use radius, perpendicular height and compatible units in the appropriate formula.
- Split composite figures or subtract cutouts, without counting shared interior edges.
- Arc length is a fraction of the full circumference.
- Cones and pyramids have one-third the volume of matching cylinders and prisms.
What you’ll be able to do
- Distinguish boundary length, flat area, and three-dimensional volume.
- Calculate perimeter, circumference, arc length, and areas of common figures.
- Find volumes of prisms, cylinders, cones, pyramids, and spheres.
- Decompose composite, curved, and shaded diagrams and preserve correct units.
First decide what is being measured; then mark only the dimensions that belong in the formula.
Name the quantity first
Perimeter is distance around a flat figure; circumference is a circle's perimeter. Both use linear units. Area covers a flat region and uses square units. Volume fills a solid and uses cubic units.
Scroll the table sideways if needed.
| Clue | Measure | Units |
|---|---|---|
| border, fence, trim | perimeter/circumference | linear |
| cover, tile, shaded region | area | square |
| fill, capacity, space inside | volume | cubic |
Use this routine: name the quantity, copy the formula, label dimensions, substitute, calculate, and attach units.
Use a perpendicular height
The height used in A = b × h ÷ 2 is the perpendicular distance to the base, not a sloping side. A triangle covers half the area of a matching base-by-height rectangle.
Use a perpendicular height — Denny Burzynski and Wade Ellis, Jr. / OpenStax CNX, via Mathematics LibreTexts. CC BY. Source image reproduced unchanged.
Trace straight and curved boundaries
Polygon perimeter is the sum of all outside sides. A square has P=4s; a rectangle has P=2l+2w. Infer missing aligned lengths when needed, and never count a shared interior segment.
For a circle, C=2πr=πd and d=2r. An arc is a fraction of the circumference: arc length=(central angle/360°)(2πr).
Example: A 90° arc in a circle of radius 8 cm is (90/360)(2π·8)=4π cm. The perimeter of a quarter-circle region also includes both radii, so it is 4π+16 cm.
Use perpendicular height for area
Scroll the table sideways if needed.
| Shape | Area | Watch for |
|---|---|---|
| square | s² | side times itself |
| rectangle | lw | perpendicular dimensions |
| triangle | bh/2 | halve the product |
| parallelogram | bh | not the slanted side |
| trapezoid | (b₁+b₂)h/2 | bases are parallel |
| rhombus | d₁d₂/2 | use diagonals |
| circle | πr² | square radius, not diameter |
Example: For trapezoid bases 8 m and 14 m with height 5 m, A=(8+14)(5)/2=55 m². A slanted leg is not the height.
Decompose composite and shaded regions
Split a composite figure into nonoverlapping familiar shapes and add. For a cutout or shaded difference, subtract the removed area from the outer area.
Example: Area=12·8+(1/2)π(4²)=96+8π m². Outside perimeter=12+12+8+4π=32+4π m. For a shaded ring, subtract circles: πR²−πr².
Build prism and cylinder volume from a base
Every prism uses V=Bh, where B is one base's area and h is the perpendicular distance between bases. Thus a rectangular prism uses V=lwh. For a triangular prism, find B=bh/2, then multiply by prism length. A cylinder uses V=πr²h.
Example: A triangular prism has a triangular base 6 cm wide and 4 cm high and is 9 cm long. B=6·4/2=12 cm², so V=12·9=108 cm³.
Volume fills three dimensions
This box holds 4 × 2 × 1 = 8 cubic inches. Multiplying only 4 × 2 gives its base area, 8 square inches, not its volume; the units distinguish the quantities even when their numerical values match.
Volume fills three dimensions — Denny Burzynski and Wade Ellis, Jr. / OpenStax CNX, via Mathematics LibreTexts. CC BY. Source image reproduced unchanged.
Recognize pointed solids and spheres
A cone uses V=(1/3)πr²h. A pyramid uses V=(1/3)Bh. Each is one-third of a matching cylinder or prism because it tapers to a point. Use perpendicular height, not slant height. A sphere uses V=(4/3)πr³.
Example: For a cone with r=4 cm and h=6 cm, V=(1/3)π(4²)(6)=32π cm³. For a rectangular pyramid with a 9 m by 5 m base and height 8 m, V=(1/3)(45)(8)=120 m³.
Check dimensions and plausibility
Mark each value as a side, base, perpendicular height, radius, diameter, or slant height. A cutout must reduce area, and a pointed solid must have less volume than its matching prism or cylinder.
If every length is multiplied by k, perimeter scales by k, area by k², and volume by k³. Keep units compatible and round only at the end.
Terms to remember
- perimeter
- Distance around the outside boundary of a figure.
- circumference
- The perimeter of a circle.
- radius
- Distance from a circle’s center to its boundary; half the diameter.
- perpendicular height
- Distance measured at a right angle to a base.
- composite figure
- A region made from two or more simpler shapes.
- base area
- The area of one base of a solid, represented by B.
- volume
- Space inside a solid, measured in cubic units.
Your quick summary
- Trace the requested boundary, including every exposed straight or curved part.
- Use perpendicular height in area and volume formulas.
- Add composite pieces or subtract cutouts; omit shared interior edges from perimeter.
- Prisms and cylinders use Bh; cones and pyramids use Bh/3; spheres use 4πr³/3.
- Use linear, square, and cubic units correctly.
Lesson quiz
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