Understanding the Area of a Right Triangle

When you have a right triangle, like triangle ABC, with given side lengths and an angle of 90 degrees, you can easily find the area using the Pythagorean theorem and the area formula for a right triangle. This article explains the process step-by-step, providing a clear understanding of how these mathematical tools are applied.

Given Information

Let's take a look at triangle ABC with the following details:

Side AB 4 cm Side BC 5 cm Angle A 90°

Identifying the Triangle Type and Hypotenuse

Since Angle A is 90°, triangle ABC is a right triangle. In a right triangle, the side opposite the right angle is the hypotenuse. Therefore, BC is the hypotenuse, and side AB is one of the legs of the triangle.

Applying the Pythagorean Theorem

To find the length of the other leg (side AC), use the Pythagorean theorem:

a2 b2 c2

Here, c is the hypotenuse, and a and b are the legs of the triangle. Substituting the given values:

AC2 BC2 - AB2

AC2 52 - 42

AC2 25 - 16

AC2 9

AC √9

AC 3 cm

Calculating the Area of the Triangle

Now that we have all three sides of the triangle, we can use the area formula for a right triangle:

Area 0.5 × base × height

In this context, side AB is the base, and side AC is the height. Plugging in the values:

Area 0.5 × 4 cm × 3 cm

Area 0.5 × 12 cm2

Area 6 cm2

Conclusion

The area of triangle ABC is 6 square centimeters. This method can be applied to any right triangle given two side lengths to find the area of the triangle.

Additional Tips

Remember that the Pythagorean theorem is a fundamental tool for working with right triangles. It's also useful to be familiar with the area formula for right triangles, which is:

Area 0.5 × (leg 1) × (leg 2)

This formula simplifies the process of calculating the area when you know the lengths of the two legs of the right triangle.