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.

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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

1. Identify the base b and the corresponding height h.
2. Multiply: \(\color{blue}{b \times h}\).
3. Divide by 2 (or multiply by ½).
4. Include the correct units (square units).

Step-by-Step Summary

1. Write the formula: \(\color{blue}{A = \frac{1}{2} \times b \times h}\).
2. Substitute the known base and height.
3. 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

1. Find the area of a triangle with base 12 cm and height 5 cm.
2. A right triangle has legs of 7 in. And 4 in. Find its area.
3. A triangle has a base of 15 m and a height of 8 m. What is its area?
4. If the area of a triangle is 24 ft² and the base is 8 ft, what is the height?
5. A triangular sail has a base of 6 m and a height of 10 m. Find the area.
6. 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

1. \(\color{blue}{\frac{1}{2} \times 12 \times 5 = 30 \text{ cm }^{2}}\)
2. \(\color{blue}{\frac{1}{2} \times 7 \times 4 = 14 \text{ in }^{2}}\)
3. \(\color{blue}{\frac{1}{2} \times 15 \times 8 = 60 m^{2}}\)
4. \(\color{blue}{24 = \frac{1}{2} \times 8 \times h \rightarrow h = 24 \div 4 = 6 \text{ ft }}\)
5. \(\color{blue}{\frac{1}{2} \times 6 \times 10 = 30 m^{2}}\)
6. A₁ = ½×\(\color{blue}{10\times 4 = 20}\); A₂ = ½×\(\color{blue}{10\times 6 = 30}\); difference = 10 cm²

Frequently Asked Questions

What is the area of a triangle formula?

The area of a triangle is A = ½ × base × height, where the height is the perpendicular distance from the base to the opposite corner. A triangle with a base of 10 cm and a height of 7 cm has an area of ½ × 10 × 7 = 35 cm². The factor of ½ is there because a triangle is half of a parallelogram with the same base and height.

How do you find the area of a triangle with 3 sides?

Use Heron’s formula. First find the semi-perimeter s = (a + b + c) ÷ 2, then A = √(s(s − a)(s − b)(s − c)). For sides 13, 14 and 15, s = 21, so A = √(21 × 8 × 7 × 6) = √7056 = 84 square units. This is handy when no height is given.

How do you find triangle area without the height?

If you know two sides and the angle between them, use A = ½ × a × b × sin C. For sides 8 and 10 with a 30° angle between them, A = ½ × 8 × 10 × 0.5 = 20 square units. If you know all three sides instead, use Heron’s formula. For a right triangle, the two legs already act as base and height.

How do you calculate triangle area in square feet?

Measure the base and height in feet, multiply them, and divide by 2. A triangular garden bed with a base of 12 ft and a height of 9 ft covers ½ × 12 × 9 = 54 square feet. If your measurements are in inches, either convert them to feet first or divide the square-inch answer by 144, since 1 ft² = 144 in².

What is the area of a triangle in coordinate geometry?

For vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃), the area is ½ × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. For (0, 0), (4, 0) and (0, 3), that gives ½ × |0 + 4 × 3 + 0| = 6 square units. On a grid you can also count the base and height directly: here they are 4 and 3.

What is triangle area and perimeter?

Area is the flat space inside the triangle, found with ½ × base × height and written in square units. Perimeter is the distance around it, found by adding the three sides and written in plain units. A right triangle with sides 3, 4 and 5 has an area of ½ × 3 × 4 = 6 square units and a perimeter of 3 + 4 + 5 = 12 units.

Related Topics

• Triangles
• Triangle Inequality
• Area of a Parallelogram
• Area of a Trapezoid
• Polygons