Lesson 1 — Area
Big idea: Area is a positive number that measures the amount of surface inside a closed figure, and a region’s area is the sum of the areas of its nonoverlapping pieces.
Words to know
| Word | Meaning |
|---|---|
| Area | The positive number that measures the region bounded by a figure. |
| Polygonal region | The union of a polygon and its interior. |
| Congruent figures | Figures with the same size and shape. |
| Nonoverlapping parts | Pieces that do not cover any of the same surface. |
| Square unit | The unit used to measure area, such as a square inch or square mile. |
Concepts to know
- Postulate 8 (The Area Postulate): Every polygonal region has a positive number called its area such that
- congruent triangles have equal areas, and
- the area of a polygonal region equals the sum of the areas of its nonoverlapping parts.
- The symbol α (alpha) is used for “the area of.” For example, α△ABC = α△CDA when the triangles are congruent.
- A polygon is just the boundary; a polygonal region includes the inside.
- Any polygonal region can be cut into triangular regions by drawing diagonals. This is how new area formulas are built from known ones.
- The Area Postulate also makes sense for other closed figures, such as a state divided into counties.
Formulas / rules
- No new formula in this lesson.
- Area is measured in square units (in², ft², m², mi²).
- Area of a whole region = sum of the areas of its nonoverlapping parts.
- If two triangles are congruent, their areas are equal.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find the area of a figure made of pieces | Split it into nonoverlapping pieces, find each area, then add. |
| Explain why two regions have equal area | Show the triangles are congruent, then use the Area Postulate. |
| Decide whether a figure is a polygon or a region | A polygon is the boundary; a polygonal region includes the interior. |
| Compare the size of two states or maps | Compare their areas in the same square units. |
| Justify an area step | Name the Area Postulate or the “sum of nonoverlapping parts” idea. |
Watch out
- Area and perimeter are different: perimeter is a distance (units), area is surface (square units).
- Do not count overlapping pieces twice.
- A larger perimeter does not always mean a larger area.
- Keep the units square; “12 square feet” is not the same as “12 feet.”