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## Geometry for Beginners – How to Find the Area of a Triangle

Welcome to another chapter of Geometry for Beginners. Our topic today deals with finding the area of a triangle. For your best understanding of this topic, you need to be well-versed in two previous topics: (1) finding the area of a rectangle, and (2) understanding that area is measured with actual squares with resulting labels like sq. in. or ft^2. If you don’t instantly get a sense of understanding of either of these, then you need to first read the appropriate Geometry for Beginners articles and get that information “ingrained in your brain.” Geometry, like Algebra, builds on top of previous knowledge. Without the previous knowledge as a foundation, the new information will not make sense and cannot be learned effectively.

If you have the formula, in symbols and in words, for area of a rectangle in your mind, and you understand why area is labeled in square units, then we can proceed to finding the area of triangles. I want to point out that while the title says we are going to find the area of a triangle–and we are–there are actually many different triangles. Rectangles vary in some ways, but the opposite sides are ALWAYS equal and the angles are ALWAYS right angles, so one side is the base *b* and the other side is the height *h*, and the formula A = bh ALWAYS applies to rectangles. Triangles have much greater variation in shape, so we will need to consider a couple different situations. This is, however, ONLY ONE formula for area of a triangle in Geometry, so memorization should be easy. This is the good news. What will be different is the location of the height. This is the part that can be troublesome.

To develop the formula for the area of a triangle, we first need a diagram of a rectangle. Remember to draw it large enough to label the bottom with both the word “base” and the symbol *b, *and to label the perpendicular side with the word “height” and symbol *h*. Beside your diagram, write the formula for area in both words and symbols. “The area of a rectangle is base times height” and A = bh.

Now, I want you to draw one of the diagonals of the rectangle in your diagram. In a rectangle, a diagonal connects opposite corners. Can you now see that the diagonal just formed two identical triangles? For these triangles, finding the area is quite simple since each triangle is one-half of the rectangle. If the rectangle is 6 in. by 8 in., then the total area is A = bh = 6 x 8 = 48 sq. in. The area then of each triangle is 24 sq. in., and leads us to the **formula for the area of a triangle: A = 1/2 bh**. (Note: It can get confusing to always use A for area since that doesn’t specify what the figure is. To handle this, we sometimes use a little picture as a subscript. Area of a triangle could then start with A but with small triangle drawn at the lower right side of the A as a subscript number would be written. I do not have this capability here, but I hope you can picture what I mean.)

**In words: The area of a triangle is one-half the product of the base and height.” Short cut version: “The area of a triangle is one-half the base times height.” Symbol version: A = 1/2 bh.**

**Caution! Caution! Caution!** Now we get to the part where you really need to pay careful attention. Remember than all rectangles have right angles, but not all triangles have right angles. When a triangle has a right angle, one leg of the right angle can be considered the base and the other leg is the height. But what if there is no right angle?

To deal with the “no right angle” situation, I want you to look at your rectangle diagram as if your had nailed some sticks together to make your rectangle. If you have ever done anything like this you know that without some additional support pieces, the rectangle starts to lean and loses those right angles. Your rectangle begins to look more like a parallelogram with opposite sides equal and opposite angles equal. (A rectangle is actually a “special case” of a parallelogram.) CONCENTRATE ON THIS NOW. As we push on the upper corner to make our rectangle lean farther and farther to the side, the base stays the same length, but the height gets SHORTER. Our rectangle that was 6 in. by 8 in. becomes a parallelogram without a right angle that **still has a base of 8 in., but the 6 in. side is no longer the height. **The area changes as the height changes. If 6 in. is no longer the height, what is?

I trust that you remember that height is ALWAYS measured as the shortest distance down to the base and this shortest distance is measured from the upper vertex straight down to the base. There is actually no line visible yet, that represents height. What we do is “drop a perpendicular” line from the upper vertex to the base. This just means that we draw a perpendicular line segment. The length of this *new line* is the HEIGHT of the parallelogram. The formula for area still stays the same: A = bh, but we must be very careful to choose the correct length as height. Without a right angle, the height is NOT the side of the parallelogram.

Drawing in a diagonal on the parallelogram does the same thing it does in the rectangle–divides the figure into two identical triangles; so the area of the triangles is still one-half the total area. Thus, the formula for the area of ALL triangles is the same as it was before: A = 1/2 bh. Again, though, we must be careful what number we are using for height.

On your paper, draw a non-right triangle. Label the bottom side with the word “base” and symbol *b*. Locate the top vertex and draw in the perpendicular line straight down from that vertex to the base. Label this new line as “height” and *h*. For now, you will only be able to find the area if you are given the height in the problem. It takes skills we do not yet have, to figure out the height if it isn’t given to us. For now, just remember that:

**The area of a triangle is one-half the base times the height, or A = 1/2 bh.**

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