How to find out the area of a triangle

There are several ways to find the area of a triangle.

Table of Contents Show

Knowing Base and Height
Example: What is the area of this triangle?
Knowing Three Sides
Knowing Two Sides and the Included Angle
Example: Find the area of this triangle:
How to Remember
How Does it Work?
Example: Find How Much Land
Triangle area formula
How to find the area of a triangle?
Area of an equilateral triangle
What are the 3 formulas for the area of a triangle?
What are the formulas for triangles?

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Knowing Base and Height

When we know the base and height it is easy.

It is simply half of b times h

Area = 12bh

(The Triangles page explains more)

The most important thing is that the base and height are at right angles. Have a play here:

Example: What is the area of this triangle?

(Note: 12 is the height, not the length of the left-hand side)

Height = h = 12

Base = b = 20

Area = ½ bh = ½ × 20 × 12 = 120

627,723, 3132, 3133

Knowing Three Sides

There's also a formula to find the area of any triangle when we know the lengths of all three of its sides.

This can be found on the Heron's Formula page.

Knowing Two Sides and the Included Angle

When we know two sides and the included angle (SAS), there is another formula (in fact three equivalent formulas) we can use.

Depending on which sides and angles we know, the formula can be written in three ways:

Area = 12ab sin C

Area = 12bc sin A

Area = 12ca sin B

They are really the same formula, just with the sides and angle changed.

Example: Find the area of this triangle:

First of all we must decide what we know.

We know angle C = 25º, and sides a = 7 and b = 10.

So let's get going:

Area =(½)ab sin C

Put in the values we know:½ × 7 × 10 × sin(25º)

Do some calculator work:35 × 0.4226...

Area = 14.8 to one decimal place

How to Remember

Just think "abc": Area = ½ a b sin C

It is also good to remember that the angle is always between the two known sides, called the "included angle".

How Does it Work?

We start with this formula:

Area = ½ × base × height

We know the base is c, and can work out the height:

the height is b × sin A

So we get:

Area = ½ × (c) × (b × sin A)

Which can be simplified to:

Area = 12bc sin A

By changing the labels on the triangle we can also get:

Area = ½ ab sin C
Area = ½ ca sin B

One more example:

Example: Find How Much Land

Farmer Rigby owns a triangular piece of land.

The length of the fence AB is 150 m. The length of the fence BC is 231 m.

The angle between fence AB and fence BC is 123º.

How much land does Farmer Rigby own?

First of all we must decide which lengths and angles we know:

AB = c = 150 m,
BC = a = 231 m,
and angle B = 123º

So we use:

Area = 12ca sin B

Put in the values we know:½ × 150 × 231 × sin(123º) m2

Do some calculator work:17,325 × 0.838... m2

Area =14,530 m2

Farmer Rigby has 14,530 m2 of land

259, 1520, 1521, 1522,260, 1523, 2344, 2345, 3940, 3941

Created by Hanna Pamuła, PhD candidate

Reviewed by Bogna Szyk and Adena Benn

Last updated: Sep 15, 2022

This triangle area calculator can help in determining the triangle area. The basic triangle area formula needs to have a base and height given, but what if we don't have it? How can we calculate the area of a triangle with 3 sides only? The triangle area calculator is here for you. Give it a go! If you are still unsure how to find the area of a triangle, check the description below.

Triangle area formula

A triangle is one of the most basic shapes in geometry. The best known and the most straightforward formula, which almost everybody remembers from school, is:

area = 0.5 * b * h, where b is the length of the base of the triangle, and h is the height/altitude of the triangle.

However, sometimes it's hard to find the height of the triangle. In that cases, many other equations may be used, depending on what you know about the triangle:

Three sides (SSS)
If you know the lengths of all sides, use the Heron's formula:
area = 0.25 * √( (a + b + c) * (-a + b + c) * (a - b + c) * (a + b - c) )

Two sides and the angle between them (SAS)
You can calculate the area of a triangle easily from trigonometry:
area = 0.5 * a * b * sin(γ)

Two angles and a side between them (ASA)

There are different triangle area formulas versions - you can use, for example, trigonometry or law of sines to derive it:
area = a² * sin(β) * sin(γ) / (2 * sin(β + γ))

If you are looking for other formulas or calculators connected with triangles, check out this right triangle calculator, pythagorean theorem calculator, and law of cosines calculator.

How to find the area of a triangle?

Assume that we know two sides and the angle between them:

Type the first side length. It can be equal to 9 inches in our example
Enter the second triangle side. Let's choose 5 in.
Determine the angle between two known sides. For example, 30 degrees.
Watch our triangle area calculator performing all calculations for you! The area for our case is equal to 11.25 in².

Area of an equilateral triangle

To calculate the area of an equilateral triangle, you only need to have the side given:

area = a² * √3 / 4

Although we didn't make a separate calculator for the equilateral triangle area, you can quickly calculate it in this triangle area calculator. Simply use the subpart for the area of a triangle with 3 sides - as you know, every side has the same length in an equilateral triangle. It's possible to calculate that area also in the angle-side-angle or side-angle-side version - you probably remember that every angle in the equilateral triangle is equal to 60 degrees (π/3 rad).

Want more?

Hanna Pamuła, PhD candidate

30 60 90 triangle45 45 90 triangleArea of a right triangle… 15 more

What are the 3 formulas for the area of a triangle?

Area of triangle = 1/2 × side 1 × side 2 × sin(θ); when 2 sides and the included angle is known, where θ is the angle between the given two sides. Area of an equilateral triangle = (√3)/4 × side. Area of an isosceles triangle = 1/4 × b√4a2−b2 4 a 2 − b 2 ; where 'b' is the base and 'a' is the length of an equal side.