Area of Triangles
The area of a triangle is half of its base times its height: A = \frac{1}{2}bh. The base is any side you choose. The height is the straight-up distance from that base to the opposite corner. For example, a triangle with a base of 10 and a height of 6 has an area of 30 square units.
Before this lesson: You should know the area of a rectangle and how to multiply fractions and decimals.
Why a Triangle Is Half of a Rectangle
Draw a rectangle around a triangle, using the triangle’s base and height. The triangle fills exactly half of that rectangle. The rectangle’s area is b \times h, so the triangle’s area is half of that.
The same idea works for every triangle. Two copies of any triangle fit together to make a parallelogram with area b \times h.
The Area of a Triangle Formula
A = \frac{1}{2} \times b \times h \qquad \text{or} \qquad A = \frac{b \times h}{2}- b is the length of the base, the side the triangle sits on.
- h is the height. It makes a right angle with the base line.
- In a right triangle, the two sides that form the right angle can be the base and height.
- In a wide (obtuse) triangle, the height can land outside the triangle. You extend the base line to meet it.
Worked Examples
Example 1: Use the formula
A triangle has a base of 10 cm and a height of 6 cm. Find its area.
Step 1. Write the formula: A = \frac{1}{2}bh.
Step 2. Substitute: A = \frac{1}{2} \times 10 \times 6.
Step 3. Multiply: 10 \times 6 = 60, and half of 60 is 30.
Answer: The area is 30 cm².
Example 2: A right triangle
A right triangle has sides of 6 in, 8 in and 10 in. The 6 in and 8 in sides form the right angle. Find its area.
Step 1. The two sides that form the right angle are the base and height. The 10 in side is the slanted side, so you don’t use it.
Step 2. Multiply and halve: A = \frac{1}{2} \times 6 \times 8 = \frac{48}{2} = 24.
Answer: The area is 24 in².
Example 3: Height outside the triangle
A wide triangle has a base of 7 m. Its height, measured outside the triangle, is 4 m. Find its area.
Step 1. The formula is the same, even when the height is outside.
Step 2. A = \frac{1}{2} \times 7 \times 4 = \frac{28}{2} = 14.
Answer: The area is 14 m².
Example 4: Find a missing height
A triangle has an area of 45 ft² and a base of 9 ft. Find its height.
Step 1. Write the formula with what you know: 45 = \frac{1}{2} \times 9 \times h.
Step 2. Double both sides to undo the half: 90 = 9h.
Step 3. Divide by 9: h = 10.
Answer: The height is 10 ft.
Example 5: Word problem with a sail
A triangular sail has a base of 2.4 m and a height of 3.5 m. Sail fabric costs $12 per square meter. How much does the fabric cost?
Step 1. Find the area: A = \frac{1}{2} \times 2.4 \times 3.5 = \frac{8.4}{2} = 4.2 m².
Step 2. Multiply by the price: 4.2 \times 12 = 50.4.
Answer: The fabric costs $50.40.
Common Mistakes with the Area of a Triangle
| Mistake | Example of the error | Fix |
|---|---|---|
| Forgetting the one-half | A = 10 \times 6 = 60 | A triangle is half a rectangle: \frac{1}{2} \times 60 = 30. |
| Using a slanted side as the height | For the 6-8-10 triangle, \frac{1}{2} \times 8 \times 10 = 40 | Use two sides that form a right angle: \frac{1}{2} \times 6 \times 8 = 24. |
| Halving twice | \frac{1}{2} \times 5 \times \frac{1}{2} \times 8 | Take half only once: \frac{1}{2} \times 5 \times 8 = 20. |
| Not doubling to find a missing side | Area 45 and base 9 gives h = 45 \div 9 = 5 | Double first: 90 \div 9 = 10. |
| Leaving off square units | Writing the area as 30 cm | Area uses square units: 30 cm². |
Area of Triangles vs. Other Shapes
| Shape | Area formula | How it connects |
|---|---|---|
| Rectangle | A = lw | The starting point |
| Parallelogram | A = bh | A rectangle with a slanted side |
| Triangle | A = \frac{1}{2}bh | Half of a parallelogram |
| Trapezoid | A = \frac{1}{2}(b_1 + b_2)h | Covered in the next lesson |
There are other triangle formulas for when you don’t know the height. You will meet them in Geometry and Trigonometry.
Practice Problems
- Find the area of a triangle with a base of 12 m and a height of 5 m.
- Find the area of a triangle with a base of 9 in and a height of 4 in.
- Find the area of a triangle with a base of 7 cm and a height of 3 cm.
- A right triangle has sides of 9 ft, 12 ft and 15 ft. Find its area.
- A triangle has a base of \frac{1}{2} ft and a height of \frac{3}{4} ft. Find its area.
- A triangle has an area of 36 cm² and a base of 8 cm. Find its height.
- A triangle has an area of 20 m² and a height of 5 m. Find its base.
- Find the area of a triangle with a base of 6.4 in and a height of 2.5 in.
- A wide triangle has a base of 10 cm. Its height, outside the triangle, is 7 cm. Find its area.
- A triangular garden bed has a base of 15 ft and a height of 8 ft. One bag of mulch covers 12 ft². How many bags are needed?
Show answers
- \frac{1}{2} \times 12 \times 5 = 30 m²
- \frac{1}{2} \times 9 \times 4 = 18 in²
- \frac{1}{2} \times 7 \times 3 = \frac{21}{2} = 10.5 cm²
- The 9 ft and 12 ft sides form the right angle: \frac{1}{2} \times 9 \times 12 = 54 ft²
- \frac{1}{2} \times \frac{1}{2} \times \frac{3}{4} = \frac{3}{16} ft²
- 2 \times 36 = 72, and 72 \div 8 = 9 cm
- 2 \times 20 = 40, and 40 \div 5 = 8 m
- \frac{1}{2} \times 6.4 \times 2.5 = \frac{16}{2} = 8 in²
- \frac{1}{2} \times 10 \times 7 = 35 cm²
- \frac{1}{2} \times 15 \times 8 = 60 ft², and 60 \div 12 = 5. You need 5 bags.
Area of a Triangle FAQ
Why do you divide by 2 to find the area of a triangle?
A triangle is exactly half of a rectangle or parallelogram with the same base and height. So you find b \times h and take half.
What is the height of a triangle?
The height is the straight-up distance from the base to the opposite corner. It always makes a right angle with the base line, even if it lands outside the triangle.
How do you find the area of a right triangle?
Multiply the two sides that form the right angle, then divide by 2. The longest side, the hypotenuse, is not used.
Does it matter which side is the base?
No. Any side can be the base, as long as you use the height that goes with it. The area comes out the same.
Summary
- The area of a triangle is A = \frac{1}{2}bh.
- A triangle is half of a rectangle or parallelogram with the same base and height.
- The height makes a right angle with the base and can fall outside the triangle.
- In a right triangle, use the two sides that form the right angle.
- To find a missing base or height, double the area, then divide.
Previous lesson: Area of Rectangles and Parallelograms
Next lesson: Area of Trapezoids
Back to the Pre-Algebra Index