Area of a Triangle Calculator
Calculate the area of a triangle using base and height, or all three sides (Heron's formula). See also Area of Rectangle Calculator and Area of Trapezoid Calculator.
How to Calculate the Area of a Triangle
The most common method is to multiply the base by the height and divide by 2: A = ½ × base × height. The height must be perpendicular to the base. If you know all three side lengths instead, use Heron's formula: first find the semi-perimeter s = (a + b + c) / 2, then A = √(s(s−a)(s−b)(s−c)). Both methods give the same result for the same triangle.
Triangle Area Formulas
A = ½ × base × height
Heron's formula:
s = (a + b + c) / 2
A = √(s × (s − a) × (s − b) × (s − c))
Example
Find the area of a triangle with base 10 and height 8:
A = ½ × base × height
A = ½ × 10 × 8
A = 40 square units
Using Heron's formula with sides 5, 6, 7:
s = (5 + 6 + 7) / 2 = 9
A = √(9 × 4 × 3 × 2) = √216
A ≈ 14.6969 square units
Triangle Area Reference Table
| Base | Height | Area |
|---|---|---|
| 3 | 2 | 3.00 |
| 4 | 3 | 6.00 |
| 5 | 4 | 10.00 |
| 6 | 5 | 15.00 |
| 8 | 6 | 24.00 |
| 10 | 8 | 40.00 |
| 12 | 9 | 54.00 |
| 15 | 10 | 75.00 |
| 20 | 12 | 120.00 |
| 25 | 15 | 187.50 |
| 50 | 30 | 750.00 |
| 100 | 60 | 3000.00 |
Solved Examples: Area of a Triangle
Example 1: Sail Area for a Boat
A triangular sail has a base of 4 meters and a height of 6.5 meters. How much fabric is needed?
Base = 4 m, Height = 6.5 m
A = ½ × base × height
A = ½ × 4 × 6.5
A = 0.5 × 26
A = 13 square meters of sail fabric
Example 2: Triangular Roof Section
A gable roof has a triangular end with a base of 8 meters and a peak height of 3.2 meters. What area needs to be sided?
Base = 8 m, Height = 3.2 m
A = ½ × base × height
A = ½ × 8 × 3.2
A = 0.5 × 25.6
A = 12.8 square meters of siding
Example 3: Triangular Garden Plot (Heron's Formula)
A triangular garden has sides measuring 7 m, 8 m, and 9 m. Find the planting area.
a = 7, b = 8, c = 9
s = (7 + 8 + 9) / 2 = 12
A = √(s(s-a)(s-b)(s-c))
A = √(12 × 5 × 4 × 3)
A = √(720)
A ≈ 26.83 square meters
Example 4: Decorative Triangular Flag
A pennant flag is an isosceles triangle with a base of 30 cm and equal sides of 50 cm each. Find its area.
Using Heron's: a = 50, b = 50, c = 30
s = (50 + 50 + 30) / 2 = 65
A = √(65 × 15 × 15 × 35)
A = √(511,875)
A ≈ 715.54 cm² (about 0.072 m²)
Practice Questions
1. A triangular window has a base of 1.2 m and a height of 0.9 m. What is its area?
Answer: A = ½ × 1.2 × 0.9 = 0.54 m²
2. A yield traffic sign is an equilateral triangle with sides of 75 cm. Find its area.
Answer: A = (√3/4) × 75² = (√3/4) × 5625 ≈ 2,435.7 cm²
3. A triangle has sides 5 cm, 12 cm, and 13 cm. Find its area.
Answer: This is a right triangle (5² + 12² = 13²). A = ½ × 5 × 12 = 30 cm²
4. A triangular park has a base of 120 m and an area of 4,200 m². What is the height?
Answer: h = 2A/base = 2 × 4200 / 120 = 70 meters
5. A pizza slice is roughly a triangle with base 15 cm and height 20 cm. What is the area of one slice?
Answer: A = ½ × 15 × 20 = 150 cm²
6. A triangular flower bed has sides of 3 m, 4 m, and 5 m. How much mulch (in m²) is needed to cover it?
Answer: Right triangle (3-4-5), A = ½ × 3 × 4 = 6 m²
Common Mistakes When Calculating Triangle Area
Forgetting to divide by 2
The formula is A = ½ × base × height, not base × height. Without the ½, you get the area of a parallelogram (double the triangle).
Using a slant side instead of the perpendicular height
The height must be perpendicular (at 90°) to the base. Using the slant height (a side of the triangle) gives an incorrect, larger area.
Applying Heron's formula with invalid sides
Three lengths only form a valid triangle if the sum of any two sides exceeds the third. Sides 2, 3, 7 cannot form a triangle because 2 + 3 = 5 < 7.
Computing the semi-perimeter incorrectly
In Heron's formula, s = (a+b+c)/2, not (a+b+c). Forgetting to divide by 2 produces a value 4× too large under the square root.
Key Takeaways
- Basic formula: A = ½ × base × height. The height must be perpendicular to the base.
- When you know all three sides, use Heron's formula: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.
- A right triangle's area is simply ½ × leg₁ × leg₂ (the two legs serve as base and height).
- For an equilateral triangle with side s: A = (√3/4) × s².
- Any triangle's area equals exactly half of the parallelogram formed by the same base and height.
- Check that your three sides satisfy the triangle inequality (a + b > c) before using Heron's formula.
Frequently Asked Questions
What is Heron's formula?
Heron's formula calculates the area of a triangle when you know all three side lengths. First compute the semi-perimeter s = (a+b+c)/2, then A = √(s(s−a)(s−b)(s−c)). It works for any triangle.
Does the height have to be inside the triangle?
Not necessarily. For obtuse triangles, the height from the longest side falls outside the triangle. The formula A = ½bh still works — the height is the perpendicular distance from the base line to the opposite vertex.
How do I know if three sides form a valid triangle?
Three sides form a valid triangle if and only if the sum of any two sides is greater than the third side. This is called the triangle inequality theorem.
What is the area of an equilateral triangle?
For an equilateral triangle with side length s, the area is A = (√3/4) × s². For example, a triangle with side 6 has area = (√3/4) × 36 ≈ 15.588.