37
Area of composite figures
Textbook: Part 2, lesson 37, pp. 5–8
GoalFind the area of a composite figure by splitting it into parts or completing it; the area of a right triangle.
New words
equal figures · teng shakllarcomposite figure · murakkab shaklcompleting method · to‘ldirish usuliarea of a triangle · uchburchak yuzi
Explanation
Figures that coincide when one is placed on the other are called equal figures, and their areas are equal. The area of a figure equals the sum of the areas of its parts. A diagonal splits a rectangle into two equal triangles, so the area of a right triangle equals half the area of the corresponding rectangle. The area of a composite figure can be found in three ways: split it into rectangles and add their areas; split it differently and add; or complete it to a large rectangle and subtract the added part. Every method must give the same answer.
Worked examples
An L-shaped figure: an outer 10 × 6 cm rectangle with a 4 × 3 cm corner cut out. Subtracting: 10 × 6 − 4 × 3 = 60 − 12 = 48 cm². Adding: 10 × 3 + 6 × 3 = 30 + 18 = 48 cm².
A right triangle with legs 8 cm and 6 cm is half of an 8 × 6 rectangle: S = 8 × 6 : 2 = 24 cm².
Class activity
“Cut and arrange”: pupils draw an L-shape on squared paper, split it into two rectangles, calculate the area and check by counting squares (scissors only with an adult's help).
Practice
1
A 20 cm × 12 cm rectangle is split into two parts, one of area 150 cm². Find the area of the other (cm²).
90
2
Find the area (cm²) of a right triangle with legs 14 cm and 9 cm.
63
3
A 5 × 4 cm corner is cut from a 12 × 9 cm rectangle. What is the area (cm²) of the remaining figure?
88
4
Why is the area of a right triangle half the area of a rectangle?
A diagonal splits the rectangle into two equal triangles, and equal figures have equal areas.