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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 is about finding the area of a triangle. For your best understanding of this topic, you should be familiar with the previous two topics: (1) finding the area of a rectangle and (2) understanding that area is measured with real squares with resulting labels like sq. in. or pi^2. If you don’t instantly understand any of these, you should first read the appropriate articles on geometry for beginners and get this information “rooted in your brain”. Geometry, like algebra, relies on prior knowledge. Without prior knowledge as a base, new information will have no meaning and cannot be learned effectively.
If you have the formula, in symbols and words, for the area of a rectangle in your mind, and you understand why the area is labeled in square units, then we can proceed to find the area of triangles. I want to point out that even though the title says we’re going to find the area of a triangle – and we are – there are actually many different triangles out there. Rectangles vary in some ways, but opposite sides are ALWAYS equal and angles are ALWAYS right angles, so one side is the base b and the other side is height h, and the formula A = bh ALWAYS applies to rectangles. Triangles are much more variable in shape, so we’ll need to consider a few different situations. This is, however, ONLY ONE formula for the area of a triangle in geometry, so memorization should be easy. This is the good news. What will be different is the height location. This is the part that can be annoying.
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 label the perpendicular side with the word “height” and the symbol h. Next to your diagram, write the area formula in words and symbols. “The area of a rectangle is the base multiplied by the 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 has just formed two identical triangles? For these triangles, finding the area is quite simple since each triangle is half of the rectangle. If the rectangle measures 6″ by 8″, then the total area is A = bh = 6 x 8 = 48 in2. The area of each triangle is then 24 in2 and brings us to the triangle area formula: A = 1/2 bh. (Note: It can be confusing to always use A for area as it doesn’t specify what the figure is. To handle this we sometimes use a small picture as a hint. The area of a triangle could then start by A but with small triangles drawn on the bottom right of the A as a subscript number would be written. I don’t have that ability here, but hope you can figure out what I mean.)
In short: the area of a triangle is half the product of the base and the height. .
Caution! Caution! Caution! Now we come to the part where you really have to be very careful. Remember that 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 you had nailed some sticks together to make your rectangle. If you’ve ever done anything like this, you know that without a few extra support pieces, the rectangle starts to lean and loses those right angles. Your rectangle starts to look more like a parallelogram with equal opposite sides and equal opposite angles. (A rectangle is actually a “special case” of a parallelogram.) FOCUS ON THIS NOW. As we push on the top corner to make our rectangle lean more and more to the side, the base stays the same length, but the height gets SHORTER. Our rectangle which measured 6 inches by 8 inches becomes a parallelogram without a right angle which still has an 8″ base, but the 6″ side is no longer the height. The area changes as the pitch changes. If 6″ isn’t the height anymore, what is it?
I hope you remember that height is ALWAYS measured as the shortest distance to the base, and that shortest distance is measured from the top vertex to the base. There is actually no visible line yet, which represents the height. What we’re doing is “dropping a perpendicular line” from the top vertex to the base. It just means that we draw a perpendicular line segment. The length of this new line is the HEIGHT of the parallelogram. The formula for the area always remains the same: A = bh, but we have to be very careful to choose the right length as the height. Without a right angle, the height is NOT the side of the parallelogram.
Drawing diagonally on the parallelogram does the same thing as in the rectangle – divides the figure into two identical triangles; so the area of the triangles is always half of the total area. So the formula for the area of ALL triangles is the same as before: A = 1/2 bh. Again, however, we have to be careful about the number we use for the height.
On your paper, draw a non-right triangle. Label the bottom side with the word “base” and the symbol b. Locate the top vertex and draw the perpendicular line 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 of the problem. It takes skills that we don’t yet have to determine the height if it is not given to us. For now, just remember that:
The area of a triangle is equal to half the base multiplied by the height, or A = 1/2 bh.
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