Depending on whether you're studying the area of a triangle, the Pythagorean Theorem, or advanced trigonometry in your high school math class, there are many ways to find the base of a triangle. Here are a couple of the most common scenarios and methods:
1. How To Find the Base from the Area of a Triangle
The formula for the area of a triangle is:
If you know the area of a triangle and also the height of the triangle, you can apply the area formula in reverse to calculate the length of the base, using algebra to isolate the variable that you care about:
2. How To Find the Base of a Right Triangle
When it comes to finding the base for a right triangle, you can apply either the Pythagorean Theorem or the area formula depending on whether you know the lengths of two sides or only the area and height. If you know the length of the hypotenuse and the other side length as well, you can apply the Pythagorean Theorem in reverse:
If you don't know the area but you know the length of the side of the triangle, you can safely use the area formula. The right angle means the height (an imaginary perpendicular line from the base) and the side of the triangle are one and the same. Consider the following triangle abc:
Here, we flip the area formula and solve:
3. How To Find the Base of an Isosceles Triangle
If you know the side length and height of a triangle that is isosceles, you can find the base of the triangle using this formula:
where the term a is the length of the two known sides of the isosceles that are equivalent.
4. Base of an Equilateral Triangle
All three sides of a triangle that is equilateral are the same length. We can use this to our advantage by once again, substituting the area formula:
To find h, we visualize the equilateral triangle as two smaller right triangles, where the hypotenuse is the same length as the side length b. By the 30-60-90 rule, a special case of a right triangle, we know that the base of this smaller right triangle is and the height of this smaller right triangle is , assuming b to be the hypotenuse.
Now that we know the height, we can apply the area formula:
And substitute for the height:
How To Find the Base of a Triangle
As you can see, the base of a triangle can be calculated through many different ways depending on the type of triangle and information you have. However, the length of the base is always related to the area of the triangle with the area formula.
The Pythagorean Theorem can help you out if you are working with right triangles. And don’t forget that the definitions of an isosceles or equilateral triangle also are tools to help you work through geometric problems like these!