How to find the base of a scalene triangle

\(\normalsize Scalene\ triangle\\(1)\ height:\hspace{10px}h=b\cdot \sin C=c\cdot \sin B\\(2)\ angle:\hspace{12px}B = \sin^{\small-1}{\large\frac{h}{c}}\ ,\hspace{20px}C = \sin^{\small-1}{\large\frac{h}{b}}\\(3)\ side:\hspace{25px} c=\sqrt{a^2+b^2-2ab\cdot \cos C}\\(4)\ area:\hspace{25px}S={\large\frac{1}{2}}ah={\large\frac{1}{2}}ab\cdot \sin C\\\hspace{100px}={\large\frac{1}{2}}a^2{\large\frac{\sin B \sin C}{\sin(B+C)}}\\\hspace{100px}=\sqrt{s(s-a)(s-b)(s-c)}\\\hspace{200px}s={\large\frac{a+b+c}{2}}\\

Índice Show

1. How To Find the Base from the Area of a Triangle
2. How To Find the Base of a Right Triangle
3. How To Find the Base of an Isosceles Triangle
4. Base of an Equilateral Triangle
How To Find the Base of a Triangle

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Updated November 03, 2020

By Chris Deziel

Unlike an equilateral triangle with its three equal sides and angles, an isosceles one with its two equal sides, or a right triangle with its 90-degree angle, a scalene triangle has three sides of random lengths and three random angles. If you want to know its area, you need to make a couple of measurements. If you can measure the length of one side and the perpendicular distance of that side to the opposing angle, you have enough information to calculate area. It's also possible to calculate area if you know the lengths of all three sides. Determining the value of one of the angles as well as the lengths of the two sides that form it also allows you to calculate area.

The area of a scalene triangle with base b and height h is given by 1/2 bh. If you know the lengths of all three sides, you can calculate area using Heron's Formula without having to find the height. If you know the value of an angle and the lengths of the two sides that form it, you can find the length of the third side using the Law of Cosines and then use Heron's Formula to calculate area.

Consider a random triangle. It's possible to scribe a rectangle around it that uses one of the sides as its base (it doesn't matter which one) and just touches the apex of the third angle. The length of this rectangle equals the length of the side of the triangle that forms it, which is called the base (b). Its width is equal to the perpendicular distance from the base to the apex, which is called height (h) of the triangle.

The area of the rectangle you just drew equals b × h. However, if you examine the lines of the triangle, you'll see they divide the pair of rectangles created by the perpendicular line from the base to the apex exactly in half. Thus, the area inside the triangle is exactly half that outside it, or 1/2 bh. For any triangle:

\text{Area} = \frac{1}{2} \text{ base} × \text{height}

Mathematicians have known how to calculate the area of a triangle with three known sides for millennia. They use Heron's Formula, named after Heron of Alexandria. To use this formula, you first have to find the half-perimeter (s) of the triangle, which you do by adding all three sides and dividing the result by two. For a triangle with sides a, b and c, the half-perimeter

s = \frac{1}{2}(a + b + c)

Once you know s, you calculate area using this formula:

\text{Area} = \sqrt{s (s - a) (s - b) (s - c)}

Consider a triangle with three angles A, B and C. The lengths of the three sides are a, b and c. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. If you know one of the angles – for example, angle C – and the two sides that form it – in this case, a and b – you can calculate the length of the third side using this formula:

c^2 = a^2 + b^2 − 2ab \cos(C)

Once you know the value of c, you can calculate area using Heron's Formula.

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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!