23
Area of a triangle
Textbook: pp. 79–81
GoalKnow the formula for the area of a triangle and its consequences, and use them.
New words
area of a triangle · uchburchak yuzibase and altitude · asos va balandlikmedian · medianaleg · katet
Explanation
Theorem: the area of a triangle is half the product of a base and its altitude: S = ½ · a · h. Proof: complete the triangle to a parallelogram; a diagonal cuts the parallelogram into two congruent triangles, so 2S = ah. Consequences: the area of a right triangle is half the product of its legs; for equal bases the areas are in the ratio of the altitudes, and for equal altitudes in the ratio of the bases; triangles with equal bases and equal altitudes have equal area. So the median of a triangle divides it into two triangles of equal area.
Worked examples
The legs of a right triangle are 9 cm and 12 cm. Its area is 9 · 12 : 2 = 54 cm².
A triangle has area 70 cm². A median splits it into two triangles of equal area, each 35 cm².
Class activity
“Half”: students draw a rectangle, cut it by a diagonal into two triangles and compare their areas.
Practice
1
A triangle has base 16 cm and altitude 9 cm. What is its area (cm²)?
72
2
The legs of a right triangle are 7 cm and 10 cm. What is its area (cm²)?
35
3
A triangle has area 60 cm² and base 15 cm. How long is the altitude to that base (cm)?
8
4
Why do triangles with equal bases and equal altitudes have equal area?
Both areas are ½ah with the same a and h, so they are equal.