26
Area of a polygon
Textbook: pp. 87–88
GoalKnow how to find the area of a polygon by splitting it into triangles or simple figures.
New words
polygon · ko‘pburchakdiagonal · diagonalsplitting into triangles · uchburchaklarga ajratishsum of areas · yuzlar yig‘indisi
Explanation
To find the area of a polygon we split it into parts whose areas are easy to compute. A convex n-gon can be split into n − 2 triangles by the diagonals from one vertex. By Property 2 of area, the area of the polygon is the sum of the areas of these triangles. Sometimes it is more convenient to split a figure into rectangles, trapezoids or triangles. Another way is to complete the figure to a large rectangle and subtract the area of the extra parts.
Worked examples
A pentagon is split from one vertex into 3 triangles of areas 12, 20 and 8. The pentagon’s area is 12 + 20 + 8 = 40.
An L-shaped figure is a 9 × 5 rectangle with a 3 × 2 corner removed. Its area is 45 − 6 = 39.
Class activity
“Split the figure”: draw a pentagon on squared paper, split it into triangles in two different ways and check that the area sum is the same both times.
Practice
1
Into how many triangles can a hexagon be split by the diagonals from one vertex?
4
2
A polygon is split into triangles of areas 7, 11 and 13. What is its area?
31
3
A 5 × 4 part is cut out of a 12 × 8 rectangle. What is the area of the remaining figure?
76
4
Why is the area of a polygon equal to the sum of the areas of the triangles it is split into?
By Property 2 of area: if the parts do not overlap, the area of the whole is the sum of the areas of the parts.