Finding the Area of a Triangle Using One Height and Two Sides

Calculating the area of a triangle can be done in various ways depending on the information available. In this article, we will explore how to find the area of a triangle when you have one height and two other sides. This method involves the use of trigonometry and Heron's formula for a more accurate result.

Method 1: Using a Base and Height

The simplest formula for finding the area of a triangle is:

Area 1/2 x base x height

This formula is applied when you know the base and height of the triangle. However, in the context of this article, we will be dealing with a scenario where the base is not given, but we have the height and two other sides. Let's explore the steps to find the missing base using trigonometry.

Step-by-Step Guide: Using Trigonometry

Here is how you can use trigonometry to find the base of the triangle:

Identify the sides and heights you have. Let's assume you have side AB 8 cm, side AC 7 cm, and the height AD 6 cm. Use the Pythagorean theorem to find the base BC. The Pythagorean theorem states:

Base^2 AB^2 - AD^2 or AC^2 - AD^2

Calculate the two possible values for the base using the Pythagorean theorem. Use the value that fits your triangle's configuration as the base. Once you have the base, use the area formula:

Area 1/2 x base x height

Let's calculate the area for the given example:

BC (8^2 - 6^2)^0.5 (7^2 - 6^2)^0.5

Substituting the values, we get:

BC (64 - 36)^0.5 (49 - 36)^0.5

BC 28^0.5 13^0.5 ≈ 5.29 3.61 8.9 cm

Area of triangle ABC (8.9 * 6) / 2 ≈ 26.7 cm^2

Method 2: Using Heron's Formula

If you know all three sides of the triangle, you can use Heron's formula to find the area. Heron's formula states:

Area sqrt[s(s - a)(s - b)(s - c)]

where s is the semi-perimeter of the triangle.

s (a b c) / 2

Let's apply Heron's formula to our example:

A sqrt[s(s - a)(s - b)(s - c)]

Here, a, b, and c are the sides of the triangle. For our example, let's assume a 8 cm, b 7 cm, and c 8.9 cm.

s (8 7 8.9) / 2 12.45

A sqrt[12.45(12.45 - 8)(12.45 - 7)(12.45 - 8.9)]

A ≈ sqrt[12.45 * 4.45 * 5.45 * 3.55] ≈ 26.7 cm^2

Conclusion

Whether you need to use trigonometry or Heron's formula, you can find the area of a triangle when you have one height and two other sides. The key is to identify the base correctly and then apply the appropriate formula. This article has provided a step-by-step guide on how to do this, ensuring you get an accurate result.

Keywords

area of triangle, height and sides, Heron's formula