Learning Objectives
- Find the area of a triangle using Heron's formula
- Apply Heron's formula to find areas of quadrilaterals by dividing into triangles
- Understand the concept of semi-perimeter
- Solve real-life problems involving areas of triangular and quadrilateral shapes
Key Concepts
Area of Triangle — Standard Formula
For a triangle with base b and height h:
Area = (1/2) × base × height
This formula requires the height, which is not always easy to determine.
Heron's Formula
When all three sides of a triangle are known but the height is not, Heron's formula is used.
For a triangle with sides a, b, and c:
Semi-perimeter: s = (a + b + c) / 2
Area = √[s(s - a)(s - b)(s - c)]
This formula works for all types of triangles — scalene, isosceles, and equilateral.
Application to Equilateral Triangle
For an equilateral triangle with side a:
s = 3a/2
Area = (√3/4) × a² (derived from Heron's formula)
Application to Isosceles Triangle
For an isosceles triangle with equal sides a and base b:
s = (2a + b)/2
Area = (b/4) × √(4a² - b²)
Area of Quadrilaterals
To find the area of a quadrilateral using Heron's formula, divide it into two triangles by drawing a diagonal. Calculate the area of each triangle using Heron's formula, then add the two areas.
If the diagonal and the lengths of all four sides are known, apply Heron's formula to each triangle separately.
Summary
Heron's formula provides a way to calculate the area of a triangle when all three sides are known, without needing the height. It uses the semi-perimeter s = (a + b + c)/2 and gives Area = √[s(s-a)(s-b)(s-c)]. This formula can be extended to find areas of quadrilaterals by splitting them into triangles using a diagonal.
Important Terms
- Semi-perimeter (s): Half of the perimeter of the triangle, s = (a + b + c)/2
- Heron's Formula: Area = √[s(s-a)(s-b)(s-c)] where s is the semi-perimeter
- Scalene Triangle: Triangle with all three sides of different lengths
Quick Revision
- Semi-perimeter s = (a + b + c)/2
- Heron's Formula: Area = √[s(s-a)(s-b)(s-c)]
- Equilateral triangle area = (√3/4) × a²
- For quadrilaterals: divide into two triangles using a diagonal
- Heron's formula works for any triangle — no height needed