Area of a Triangle
The area of a triangle is one of the most useful formulas in geometry and a guaranteed topic on the GED Math test. Whether you are working with a right triangle on a grid or a scalene triangle in a word problem, one simple formula — half of base times height — handles every case.
What Is the Area of a Triangle?
The area of a shape is the amount of flat surface it covers, measured in square units. For a triangle, the area is always half the area of a parallelogram with the same base and height. That is why the formula has a factor of ½.
A = ½ × \(\color{blue}{b \times h}\)
- b = base (any one side of the triangle)
- h = height (the perpendicular distance from the base to the opposite vertex)
How to Use the Formula
Identifying the base and height
Any side of the triangle can be the base. The height is always drawn perpendicular (at 90°) to the base. In a right triangle, the two legs are base and height. In other triangles, the height may be inside or outside the triangle.
Applying the formula
- Identify the base b and the corresponding height h.
- Multiply: \(\color{blue}{b \times h}\).
- Divide by 2 (or multiply by ½).
- Include the correct units (square units).
Step-by-Step Summary
- Write the formula: \(\color{blue}{A = \frac{1}{2} \times b \times h}\).
- Substitute the known base and height.
- Compute and label with square units.
Watch: Finding the Area of a Triangle (Video Lesson)
Math with Mr. J provides a clear, step-by-step guide to the triangle area formula:
Worked Examples
Example 1: Find the area of a triangle with base 6 cm and height 4 cm.
\(\color{blue}{A = \frac{1}{2} \times 6 \times 4 = \frac{1}{2} \times 24 = 12 \text{ cm }^{2}}\)
Example 2: A right triangle has legs of length 10 ft and 5 ft. Find its area.
The two legs serve as base and height.
\(\color{blue}{A = \frac{1}{2} \times 10 \times 5 = \frac{1}{2} \times 50 = 25 \text{ ft }^{2}}\)
Example 3: A triangle has a base of 8 m and a height of 3 m. Find the area.
\(\color{blue}{A = \frac{1}{2} \times 8 \times 3 = \frac{1}{2} \times 24 = 12 m^{2}}\)
Example 4: A triangular garden has a base of 9 yards and a height of 6 yards. How many square yards of sod are needed to cover it?
\(\color{blue}{A = \frac{1}{2} \times 9 \times 6 = \frac{1}{2} \times 54 = 27 \text{ yd }^{2}}\)
More Practice: Area of Triangles Intuition (Video)
Khan Academy explains why the formula is half of base times height:
Exercises
- Find the area of a triangle with base 12 cm and height 5 cm.
- A right triangle has legs of 7 in. and 4 in. Find its area.
- A triangle has a base of 15 m and a height of 8 m. What is its area?
- If the area of a triangle is 24 ft² and the base is 8 ft, what is the height?
- A triangular sail has a base of 6 m and a height of 10 m. Find the area.
- Two triangles have the same base of 10 cm. One has height 4 cm; the other has height 6 cm. How much larger is the second triangle’s area?
Answers
- \(\color{blue}{\frac{1}{2} \times 12 \times 5 = 30 \text{ cm }^{2}}\)
- \(\color{blue}{\frac{1}{2} \times 7 \times 4 = 14 \text{ in }^{2}}\)
- \(\color{blue}{\frac{1}{2} \times 15 \times 8 = 60 m^{2}}\)
- \(\color{blue}{24 = \frac{1}{2} \times 8 \times h \rightarrow h = 24 \div 4 = 6 \text{ ft }}\)
- \(\color{blue}{\frac{1}{2} \times 6 \times 10 = 30 m^{2}}\)
- A1 = ½×\(\color{blue}{10\times 4 = 20}\); A2 = ½×\(\color{blue}{10\times 6 = 30}\); difference = 10 cm²
Frequently Asked Questions
Why is the area of a triangle half the area of a parallelogram?
Any triangle can be doubled by attaching an identical copy (rotated 180°) to form a parallelogram with the same base and height. So the triangle’s area is exactly half the parallelogram’s area \(\color{blue}{(b \times h)}\).
What if the height is not given but falls outside the triangle?
For obtuse triangles, the height drawn to the base may fall outside the triangle. The formula \(\color{blue}{A = \frac{1}{2}\text{ bh }}\) still applies — you just use that external perpendicular distance as h.
What units do I use for area?
Area is always expressed in square units: cm², ft², m², in², etc. If the base is in inches and the height is in inches, the area is in square inches.
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