Area of Parallelograms and Triangles

Finding the area of parallelograms and triangles is a fundamental skill in geometry. Whether you are measuring a slanted roof or a triangular piece of land, all you need to know is the shape's base and height.

The Golden Rule: Base and Height

Before you calculate the area of either shape, you must identify the base ($b$) and the height ($h$).

The most important rule to remember is that the height must always be perpendicular (at a 90-degree angle) to the base. The height is usually shown with a small square symbol representing a right angle. Never use a slanted side as your height!

Area of a Parallelogram

A parallelogram is a four-sided shape where opposite sides are parallel. Think of it as a slanted rectangle. If you were to cut off the slanted triangular piece from one side and move it to the other, it would form a perfect rectangle. Because of this, it uses the exact same area formula as a rectangle.

Formula: $A = b \times h$

Example 1: Finding the Area

Question: Find the area of a parallelogram with a base of $8\text{ cm}$ and a height of $5\text{ cm}$.

Solution: $A = 8 \times 5 = 40\text{ cm}^2$ The area is $40\text{ square centimeters}$.

Example 2: Finding a Missing Dimension

Question: A parallelogram has an area of $48\text{ cm}^2$ and a height of $6\text{ cm}$. What is its base?

Solution: Set up the formula with the numbers you know: $48 = b \times 6$ To find the base, divide the area by the height: $b = 48 \div 6 = 8\text{ cm}$ The base is $8\text{ cm}$.

Area of a Triangle

If you take any parallelogram and draw a diagonal line from one corner to the opposite corner, you will cut it into two identical triangles. Because of this relationship, the area of a triangle is exactly half the area of a parallelogram that has the same base and height.

Formula: $A = \frac{1}{2} \times b \times h$

Example 3: Finding the Area of a Triangle

Question: Find the area of a triangle with a base of $12\text{ m}$ and a height of $7\text{ m}$.

Solution: First, multiply the base by the height: $12 \times 7 = 84$ Then, take half of that result: $A = \frac{1}{2} \times 84 = 42\text{ m}^2$ The area is $42\text{ square meters}$.

Summary

• Parallelogram: Multiply base times height ($A = b \times h$).
• Triangle: Multiply base times height, then divide by 2 ($A = \frac{1}{2} \times b \times h$).
• Always ensure the height you use forms a right angle with the base!