Area of a Triangle
In short: The area of a triangle is found by multiplying a base by its matching height and halving the product, since two identical triangles tile a parallelogram.
6th GradeGeometry6.G.A.1
What is Area of a Triangle?
Finding the area of a triangle starts with a picture: build a rectangle or parallelogram on the same base and height, and the triangle fills exactly half of it. That is where the one half in the formula comes from.
Precisely, \(A = \frac{1}{2} b h\), where \(b\) is any side chosen as the base and \(h\) is the perpendicular distance from that base to the opposite vertex. The height is not a slanted side unless the triangle has a right angle there.
How to spot it
- A problem gives a triangular sail, flag, roof end or garden bed and asks how much surface it covers.
- A drawing shows a dashed line meeting a side at a right angle: that is the height.
- The question lists a base and a height, with a third slanted side as a distraction.
- A polygon is cut into triangles so that its area can be added up.
The method
- Choose a side as the base and find the matching height, drawn at 90 degrees to that base.
- Multiply base by height.
- Take half of the product, by dividing by 2 or multiplying by 0.5.
- Write the area of a triangle in square units and check that it is half of the surrounding rectangle.
Area of a Triangle: solved example
Problem
A sailing club in San Diego cuts a triangular training sail with a bottom edge of 6.5 feet and a perpendicular height of 12 feet. How much cloth covers one side?
Solution
Use \(A = \frac{1}{2} b h\) with \(b = 6.5\) and \(h = 12\).
\(\frac{1}{2} \cdot 6.5 \cdot 12 = 0.5 \cdot 78 = 39\). The sail has 39 square feet of cloth.
Try it
Question 1
A triangle has base 18 inches and height 7 inches. Find its area.
Show the answer
\(\frac{1}{2} \cdot 18 \cdot 7 = 63\) square inches.
Question 2
A triangle has area 54 square meters and base 12 meters. Find its height.
Show the answer
\(54 = \frac{1}{2} \cdot 12 \cdot h = 6h\), so \(h = 9\) meters.
Area of a Triangle vs. Area of a Parallelogram
Give a slanted four-sided shape the same base and height and it covers double: \(A = bh\). Only the three-sided rule carries the factor \(\frac{1}{2}\).
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