19
The idea of area. Figures of equal area
Textbook: pp. 69–71
GoalKnow the idea of area, the basic properties of area, and figures of equal area and equidecomposable figures.
New words
area · yuzfigures of equal area · tengdosh shakllarequidecomposable figures · teng tuzilgan shakllarsimple figure · sodda shakl
Explanation
A figure that can be cut into finitely many triangles is called a simple figure; every convex polygon is one. Area is a positive quantity with two basic properties. Property 1: congruent figures have equal areas. Property 2: if a polygon is made of non-overlapping polygons, its area is the sum of their areas. If one polygon can be cut into parts that are rearranged into another, the polygons are called equidecomposable. Polygons with equal areas are called figures of equal area: congruent figures have equal area, but figures of equal area are not necessarily congruent.
Worked examples
Rectangles 2 × 6 and 3 × 4 have equal areas (12 and 12), but they are not congruent: their sides differ.
A figure consists of two non-overlapping rectangles, 3 × 4 and 2 × 5. Its area is 12 + 10 = 22 square units.
Class activity
“Find equal areas”: on squared paper draw different rectangles of area 12 squares and compare them.
Practice
1
A figure consists of non-overlapping rectangles 5 × 2 and 3 × 4. What is its area in square units?
22
2
Are the rectangles 6 × 4 and 8 × 3 of equal area? Are they congruent?
They have equal area (both 24), but they are not congruent.
3
A parallelogram is made of two congruent triangles of area 15 square units each. What is its area?
30
4
Why need figures of equal area not be congruent?
For example, rectangles 2 × 6 and 3 × 4 have equal area but different sides.