Key takeaway: The area of a triangle is found by halving the product of a chosen side and the straight-line distance from that side to the opposite corner, measured at a right angle.
Grade 66.G.A.1Geometry and Measurement
Definition of Area of a Triangle
The area of a triangle measures the flat region enclosed by its three sides. Put two identical triangles together, one rotated half a turn, and they form a parallelogram. Each triangle is therefore exactly half of a parallelogram that shares its base and its height.
Whichever side you pick plays the role of base. The matching height is the shortest trip from that side, or from its extension, up to the opposite corner, taken at a right angle. In a slanted triangle the height is drawn as a dashed segment inside the figure, and it is usually shorter than the slanted sides, so be careful which number you pick.
Formula and notation
Using words for the two measurements that matter:
\[A = \frac{\text{base} \times \text{height}}{2}\]
The base and the height must meet at a right angle and use the same unit; the result is in square units.
Key facts
- The area of a triangle is half the surface of a matching parallelogram built on the same side.
- The height meets the base at a right angle and need not lie along a side.
- Any side may serve as the base, if you use the matching height.
- Side lengths that are not perpendicular to the base are never used in the formula.
Area of a Triangle example
A triangular racing sail has a foot (bottom edge) of 18 feet. Straight up from that edge to the top corner it measures 7.5 feet, while the slanted edge on the left runs 9.5 feet. How much cloth does the sail contain?
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The bottom edge is the base: 18 feet.
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The straight-up distance is the height: 7.5 feet. The 9.5-foot edge is slanted, so it is ignored.
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Compute: \(\frac{1}{2} \times 18 \times 7.5 = 9 \times 7.5 = 67.5\).
The sail has an area of 67.5 square feet.
Common mistake
Watch out
Evan computes \(\frac{1}{2} \times 18 \times 9.5 = 85.5\) square feet for the sail.
Why it is wrong
He used the slanted side as the height. A height must make a right angle with the base, and here that distance is 7.5 feet. A slanted side is longer than the true height, so it overestimates the area of a triangle.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Geometry and Measurement