Area of a Triangle

1. What is the Area of a Triangle?

The area of a triangle is the total region enclosed by its three sides, measured in square units. In other words, it is the amount of two-dimensional space that lies inside the triangle.

Area is always expressed in square units — for example, cm², m², in², or ft² — because we are measuring a two-dimensional surface. The basic formula uses two measurements: the base (any side of the triangle) and the height (the perpendicular distance from that base to the opposite vertex).

A  =  ½  ×  base  ×  height

Where A = area (in square units), base = length of the chosen side, height = perpendicular height to that side.

2. Area of a Triangle Formula

The most commonly used formula is A = ½ × b × h, where:

• A = area of the triangle (cm², m², in², ft², …)
• b = base — any side you choose as the reference
• h = height — the perpendicular distance from the base to the opposite vertex

This formula works for all types of triangles — right triangles, acute triangles, obtuse triangles, equilateral triangles, and isosceles triangles — as long as you use the perpendicular height, not a slanted side.

$A = \frac{1}{2} b h$

3. Why Does the Formula Have ½?

A triangle is exactly half of a parallelogram (or rectangle) with the same base and height. If you take any triangle and duplicate it, you can always fit the two copies together to form a parallelogram. Since the area of a parallelogram is base × height, the area of the triangle is half of that.

This is why every triangle area formula contains the factor ½ — it reflects this fundamental geometric relationship. Understanding this “why” helps you remember the formula rather than just memorising it.

4. How to Find the Area — Step by Step

There are 5 methods for finding the area of a triangle, depending on what information you have. Choose the method that matches your known values:

Method 1: Base and Height Known

This is the standard method. Use it whenever you know (or can measure) the base and the perpendicular height.

$A = \frac{1}{2} \times b \times h$

Example 1 — Integers: A triangle has base = 6 cm and height = 4 cm.

A = ½ × 6 × 4 = ½ × 24 = 12 cm²

Example 2 — Decimals: A triangle has base = 8.5 cm and height = 3.2 cm.

A = ½ × 8.5 × 3.2 = ½ × 27.2 = 13.6 cm²

Example 3 — Word problem: A farmer wants to fence a triangular garden. The base is 12 m and the perpendicular height is 5 m. What is the area?

A = ½ × 12 × 5 = ½ × 60 = 30 m²

Tip: The height must be perpendicular to the base. If a problem gives you a slant side instead, use Heron’s formula (Method 2) or the SAS formula (Method 3).

Method 2: Three Sides Known (Heron’s Formula)

When you know all three side lengths but not the height, use Heron’s formula. It works for any triangle.

Step 1: Calculate the semi-perimeter: $s = (a + b + c) / 2$

Step 2: Apply Heron’s formula: $A = \sqrt{ s(s – a)(s – b)(s – c) }$

Example 1: Triangle with sides a = 3, b = 4, c = 5.

s = (3 + 4 + 5) / 2 = 6

A = √( 6 × (6−3) × (6−4) × (6−5) ) = √( 6 × 3 × 2 × 1 ) = √36 = 6 cm²

Note: 3-4-5 is a right triangle, so you can verify using Method 1: A = ½ × 3 × 4 = 6 cm² ✓

Example 2: Triangle with sides a = 5, b = 12, c = 13.

s = (5 + 12 + 13) / 2 = 15

A = √( 15 × (15−5) × (15−12) × (15−13) ) = √( 15 × 10 × 3 × 2 ) = √900 = 30 cm²

Method 3: Two Sides and Included Angle (SAS)

If you know two sides and the angle between them (the included angle), use the SAS formula: $A = \frac{1}{2} \times a \times b \times \sin(C)$

Where a and b are the two known sides, and C is the included angle between them.

Example: Two sides of length a = 8 cm and b = 10 cm with an included angle C = 30°.

A = ½ × 8 × 10 × sin(30°) = ½ × 8 × 10 × 0.5 = 20 cm²

Remember: Use the sine of the included angle. Make sure your calculator is set to degrees (not radians) when working with degree angles.

Method 4: Coordinate Geometry

When a triangle is defined by three coordinate points — (x₁, y₁), (x₂, y₂), (x₃, y₃) — use the Shoelace formula:

$A = \frac{1}{2} \left| x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) \right|$

The absolute value ensures the result is always positive, regardless of the order in which you list the vertices.

Example: Triangle with vertices at A(1, 2), B(4, 6), C(7, 2).

A = ½ |1(6 − 2) + 4(2 − 2) + 7(2 − 6)|

A = ½ |1×4 + 4×0 + 7×(−4)|

A = ½ |4 + 0 − 28| = ½ × |−24| = ½ × 24 = 12 square units

Method 5: Equilateral Triangle

An equilateral triangle has all three sides equal. There is a dedicated formula derived from Heron’s formula that is faster to apply:

$A = \left(\frac{\sqrt{3}}{4}\right) \times a^2$

Where a is the length of any side.

Example: An equilateral triangle with side a = 6 cm.

$A = (\sqrt{3} / 4) \times 6^2 = (\sqrt{3} / 4) \times 36 = 9\sqrt{3} \approx 15.59 \text{ cm}^2$

5. Formula Summary Table

Use this table to quickly choose the right method for your problem:

Method	Formula	When to use	Example
1. Base & Height	A = ½ × b × h	Height is known	b=6, h=4 → A=12 cm²
2. Heron’s	A = √(s(s−a)(s−b)(s−c))	All 3 sides known, no height	3,4,5 → A=6 cm²
3. SAS	A = ½ ab sin(C)	2 sides + included angle	8,10,30° → A=20 cm²
4. Coordinates	A = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|	Vertices given as (x,y)	(1,2),(4,6),(7,2) → 12
5. Equilateral	A = (√3/4)a²	All sides equal	a=6 → A≈15.59 cm²

6. Common Mistakes to Avoid

Mistake 1: Forgetting the ½ — This is the most common error. Students often write A = b × h instead of A = ½ × b × h. Always remember: a triangle is half a parallelogram.

Mistake 2: Using a slant side as the height — The height must be the perpendicular distance from the base to the opposite vertex. If a problem labels a slant edge as the height, use Method 3 (SAS) or Heron’s formula instead.

Mistake 3: Mixing up units (cm vs cm²) — Area is always in square units. If base = 5 cm and height = 4 cm, then area = 10 cm² (not 10 cm). Including the correct unit is essential in exams.

7. Practice Problems

Try these problems yourself. Solutions are shown beneath each question.

a) A triangle has base = 10 cm and height = 6 cm. Find the area.

→ Answer: A = ½ × 10 × 6 = 30 cm²

b) A triangle has sides 7 cm, 24 cm, and 25 cm. Use Heron’s formula to find the area.

→ Answer: s = 28; A = √(28×21×4×3) = √7056 = 84 cm²

c) Two sides of a triangle are 9 m and 12 m. The included angle is 60°. Find the area.

→ Answer: A = ½ × 9 × 12 × sin(60°) = ½ × 9 × 12 × 0.866 ≈ 46.77 m²

d) A triangular plot has vertices at (0,0), (6,0), and (3,4). Find its area.

→ Answer: A = ½|0(0−4)+6(4−0)+3(0−0)| = ½|0+24+0| = 12 square units

e) (Challenge) An equilateral triangle has a perimeter of 18 cm. Find the area.

→ Answer: Side = 6 cm; A = (√3/4) × 36 = 9√3 ≈ 15.59 cm²

8. Frequently Asked Questions

9. What to Learn Next

Now that you understand how to find the area of a triangle, you are ready to explore related topics:

• Prerequisite: Types of Angles — Understanding acute, right, and obtuse angles will help you work with the SAS formula and recognise when a triangle has a 90° angle.
• Related: Types of Triangles — Learn the properties of equilateral, isosceles, scalene, right, and obtuse triangles, and how each one affects which area formula to use.
• Next step: Pythagorean Theorem — Once you can find the area, the Pythagorean Theorem lets you find missing sides in right triangles, which often feeds back into the area calculation.