Area of a Triangle- Methods and Examples
How to Find the Area of a Triangle
Finding the area of a triangle comes down to one core idea: half the base times the height. Everything else is variations on that theme. There are several methods, and the right one depends on what information you already have.
This guide covers every method you'll actually need, with examples you can follow. Skip the theory lectures—here's how it works.
The Standard Formula: Base × Height ÷ 2
Most people learn this one first. If you know the base length and the perpendicular height, you're done.
Formula: A = ½ × base × height
The height must be the perpendicular distance from the base to the opposite vertex. Not the slanted side length—always measure straight down to the base.
Example
Triangle with base = 8 cm and height = 5 cm.
A = ½ × 8 × 5 = 20 cm²
That's it. If your triangle has a right angle, the two legs are your base and height automatically. No extra work needed.
Heron's Formula: When You Know All Three Sides
Sometimes you don't have the height. You just have three side lengths. That's what Heron's formula handles.
Step 1: Find the semi-perimeter first.
s = (a + b + c) ÷ 2
Step 2: Plug it into the main formula.
A = √[s(s - a)(s - b)(s - c)]
The letter s stands for semi-perimeter. Don't confuse it with the full perimeter.
Example
Triangle with sides: a = 7, b = 9, c = 12
s = (7 + 9 + 12) ÷ 2 = 14
A = √[14(14-7)(14-9)(14-12)]
A = √[14 × 7 × 5 × 2]
A = √980
A ≈ 31.3 units²
Two Sides and the Included Angle
When you know two sides and the angle between them, use this variation:
Formula: A = ½ × a × b × sin(C)
a and b are the two known sides. C is the angle between them. This comes from trigonometry, but the math is straightforward.
Example
Sides a = 6 and b = 10, with included angle C = 30°.
A = ½ × 6 × 10 × sin(30°)
A = 30 × 0.5
A = 15 units²
You need a calculator for the sine values unless you're using common angles like 30°, 45°, or 60°.
Coordinate Geometry Method
When triangle vertices are given as coordinates, you can find the area without drawing anything.
Formula: A = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
The absolute value bars make sure you get a positive answer regardless of point order.
Example
Vertices: (2, 3), (8, 7), (5, 11)
A = ½|2(7-11) + 8(11-3) + 5(3-7)|
A = ½|2(-4) + 8(8) + 5(-4)|
A = ½|(-8) + 64 + (-20)|
A = ½|36|
A = 18 square units
Which Formula Should You Use?
Pick the method based on what you're given:
- Base and height known → Use A = ½bh. Fastest and easiest.
- All three sides known → Use Heron's formula.
- Two sides and angle known → Use A = ½ab sin(C).
- Coordinates given → Use the coordinate formula.
- Right triangle → The two legs are base and height. No height calculation needed.
Quick Comparison Table
| Method | What You Need | Complexity |
|---|---|---|
| Base × Height | Base and perpendicular height | Low |
| Heron's Formula | All three sides | Medium |
| Two sides + angle | Two sides and included angle | Medium |
| Coordinate method | Three coordinate points | Medium-High |
Common Mistakes to Avoid
- Using the slanted side as height. Height must be perpendicular to the base. Measure straight down.
- Forgetting to halve the result. The ½ in the formula isn't optional.
- Messing up Heron's semi-perimeter. Divide by 2, not the full perimeter.
- Wrong angle in trig formula. Must be the angle between the two known sides.
Getting Started: Pick Your Method
- Look at what you know. Read the problem carefully. What measurements do you have?
- Match to a formula. Use the table above to find which method fits.
- Plug in the numbers. Double-check each value before you calculate.
- Include units. If you're working in centimeters, your answer is in square centimeters (cm²).
Most textbook problems give you exactly what you need for one specific method. The hard part is recognizing which formula applies. Once you know your data, it's just arithmetic.