How To Find Area Of Triangle Without Height

How To Find the Height of a Triangle in 3 Different Situations

In trigonometry, the height of a triangle can be adamant in many different means depending on whether information technology's a right triangle, isosceles triangle (a triangle with two equal sides), or equilateral triangle.

how to find height of a triangle: what each side of a triangle is called

1. How To Observe the Height of a Right Triangle

Earlier we showtime, here's what you need to know about right triangles. A right triangle has three sides: the hypotenuse, height, and base of the triangle. The base of operations and height of a correct triangle are always the sides next to the right angle, and the hypotenuse is the longest side.

The height of a right triangle can exist determined with the surface area formula:

how to find height of a triangle: area formula

If the given area isn't known, y'all tin use the Pythagorean theorem to solve for the height of a correct triangle. Here's what the Pythagorean theorem states, given c is the hypotenuse and a and b are the other two sides:

how to find height of a triangle: Pythagorean theorem

how to find height of a triangle: diagram of a right triangle

Let's take the units from the effigy above and plug in the length of the base and hypotenuse to solve for the missing peak:

how to find height of a triangle: formula for solving the missing height

2. Finding the Height of a Not-Right Triangle

Unfortunately, you can't use the Pythagorean theorem to find the height of an isosceles triangle or the peak of an equilateral triangle (where all sides of the triangle are equal). Instead, you'll accept to draw a perpendicular line through the base of the triangle to form a right angle:

how to find height of a triangle: diagram of an equilateral and isosceles triangle

This line represents the height of these non-right triangles. Once you lot've formed this line, you'll take to use Heron'due south formula to solve for the surface area of the entire triangle.

Heron's Formula

The first step of Heron's formula is calculating half the triangle's perimeter. In this example, s represents half the perimeter and a, b, and c are the sides:

how to find height of a triangle: Heron's formula

Once you've determined s, use the post-obit formula to calculate the surface area of a triangle. Again, the two sides are a and b, and the longest side (the hypotenuse) is c:

Heron's formula

Let'due south plug in the side lengths from this isosceles triangle to observe the area of the triangle:

diagram of an isosceles triangle

formula for the area of an isosceles triangle

Now, we'll substitute s in the area formula for a non-right triangle.

formula for the area of a non-right triangle

Using Area To Discover the Meridian of a Triangle

Now that you know the surface area of the triangle pictured above, you lot can plug information technology into triangle formula A=1/2bh to find the peak of the triangle. In this instance, the base would equal half the distance of five (2.5), since this is the shortest side of the triangle.

formula to find the height of a triangle

Master Triangle Height Formulas

Finding the acme of a triangle is a multi-step process that can be confusing. However, mastering information technology helps you lot learn unlike types of area formulas, such as heron'south formula and A=1/2bh. It too shows yous how to utilise the Pythagorean theorem and triangle perimeter formulas to make up one's mind other quantities inside a triangle.