How to Calculate the Areas of Triangles and Quadrilaterals

Step-by-step Guide: Areas of Triangles and Quadrilaterals

1. Triangles: A triangle is a three-sided polygon.
– Area: The area \(A\) of a triangle with base \(b\) and height \(h\) perpendicular to the base is:
\( A = \frac{1}{2} b \times h \)

2. Quadrilaterals: A quadrilateral is a polygon with four sides.
– Area of a Rectangle: For a rectangle with length \(l\) and width \(w\), the area \(A\) is:
\( A = l \times w \)
– Area of a Square: For a square with side \(s\), the area \(A\) is:
\( A = s^2 \)
– Area of a Trapezoid: For a trapezoid with parallel sides (bases) \(a\) and \(b\), and height \(h\) perpendicular to the bases, the area \(A\) is:
\( A = \frac{1}{2} (a + b) \times h \)
– Area of a Parallelogram: For a parallelogram with base \(b\) and height \(h\) perpendicular to the base, the area \(A\) is:
\( A = b \times h \)
– Area of a Rhombus: The area \(A\) of a rhombus with diagonals \(d_1\) and \(d_2\) is:
\( A = \frac{1}{2} d_1 \times d_2 \)

Examples

Example 1:
Calculate the area of a trapezoid with bases \(7 \text{cm}\) and \(5 \text{cm}\), and a height of \(6 \text{cm}\).

Solution:
\(A = \frac{1}{2} (7 + 5) \times 6 = 36 \text{cm}^2\).

Example 2:
If a rhombus has diagonals measuring \(10 \text{cm}\) and \(8 \text{cm}\), what’s its area?

Solution:
\(A = \frac{1}{2} \times 10 \times 8 = 40 \text{cm}^2\).

Example 3:
Calculate the area of a triangle with a base of \(10 \text{cm}\) and a height of \(5 \text{cm}\).

Solution:
\(A = \frac{1}{2} \times 10 \times 5 = 25 \text{cm}^2\).

Example 4:
If the area of a triangle is \(30 \text{cm}^2\) and its base is \(10 \text{cm}\), find its height.

Solution:
Using the formula:
\(30 = \frac{1}{2} \times 10 \times h\)
Solving for (h), we get:
\(h = 6 \text{cm}\)

Practice Questions:

1. Determine the area of a triangle with a base of \(12 \text{cm}\) and a height of \(8 \text{cm}\).
2. Calculate the area of a square with a side length of \(6 \text{cm}\).
3. What is the area of a parallelogram with a base of \(9 \text{cm}\) and a height of \(5 \text{cm}\)?
4. A rhombus has one diagonal measuring \(15 \text{cm}\). If its area is \(45 \text{cm}^2\), find the length of the other diagonal.

Answers:

1. \(48 \text{cm}^2\).
2. \(36 \text{cm}^2\).
3. \(45 \text{cm}^2\).
4. \(6 \text{cm}\).