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Heron’s formula for the area of a triangle
Textbook: p. 103
GoalKnow Heron’s formula for the area of a triangle and use it to find the area from the sides.
New words
Heron’s formula · Geron formulasisemi-perimeter · yarim perimetrarea of a triangle · uchburchak yuzisquare root · ildiz
Explanation
We know the area is S = ½ · a · h_a. Substituting the expression of the altitude through the sides gives S = √(p(p − a)(p − b)(p − c)), where p = (a + b + c) : 2 is the semi-perimeter. This is called Heron’s formula; it is attributed to Heron of Alexandria, who lived in the 1st century AD. The formula is convenient when all three sides are known and no altitude is given. To compute, first find p, then p − a, p − b, p − c, multiply them and take the square root.
Worked examples
Sides 25, 25, 14: p = 32 and 32 · 7 · 7 · 18 = 28 224 = 168², so the area is 168.
Sides 7, 15, 20: p = 21 and 21 · 14 · 6 · 1 = 1764 = 42², so the area is 42.
Class activity
“Two ways to the area”: find the area of the triangle with legs 6 and 8 first from the legs, then with Heron’s formula, and compare.
Practice
1
What is the area of the triangle with sides 5, 29, 30?
72
2
What is the area of the triangle with sides 9, 10, 17?
36
3
The triangle 20, 21, 29 is right (20² + 21² = 29²). Find its area from the legs.
210
4
What makes Heron’s formula convenient?
It gives the area from the three sides alone, without finding an altitude.