A scalene triangle has three sides that have different lengths from each other and three angles that also have different measurements. There are three main methods that we can use to calculate the area of a scalene triangle depending on the information we have. We can calculate the area using the length of the base and the height. In addition, we can calculate the area using the lengths of the three sides. Also, we can calculate the area if we know the length of its two sides and the angle between those sides.

In this article, we will learn about the formulas that we can use to calculate the area of a scalene triangle using the three mentioned methods. Also, we will look at some solved examples in which we will apply these formulas to find the area with the given information.

## How to find the area of a scalene triangle?

To find the area of a scalene triangle, we need one of the following sets of measurements:

**a)** The length of one side and the perpendicular distance from that side to the opposite angle (the height).

**b)** The lengths of the three sides.

### Area of the scalene triangle with base and height

To find the area of a scalene triangle if we know the length of its base and the corresponding height, we can use the following formula:

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

where *b* is the length of the base and *h* is the length of the height.

### Area of the scalene triangle without using the height

To find the area of the scalene triangle if we know the length of two sides and the measure of the angle between them, we can use the following formula:

$latex A=\frac{ab}{2}\times \sin(C)$ |

where *a* and *b* are the lengths of two sides and C is the measurement of the angle between those sides.

### Area of the scalene triangle with the length of the three sides

To find the length of a scalene triangle if we know the length of its three sides, we can use Heron’s formula:

$latex A=\sqrt{S(S-a)(S-b)(S-c)}$ |

where, $latex a, ~b, ~c$ represent the lengths of the sides and S represents the semi perimeter that can be found with the following formula:

$latex S=\frac{a+b+c}{2}$

## Area of a scalene triangle – Examples with answers

In the following examples, we use the formulas detailed above to find the area of the scalene triangles. Each example has its solution, but it is recommended that you try to solve the exercises yourself before looking at the answer.

**EXAMPLE 1**

A scalene triangle has a base of 10 m and a height of 8 m. What is its area?

##### Solution

We can recognize the following values:

- Base, $latex b=10$ m
- Height, $latex h=8$ m

We use the first formula with these values:

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

$latex A=\frac{1}{2}\times 10\times 8$

$latex A=40$

The area is 40 m².

**EXAMPLE 2**

What is the area of a scalene triangle that has a base of 16 cm and a height of 18 cm?

##### Solution

We have the following information:

- Base, $latex b=16$ cm
- Height, $latex h=18$ cm

We substitute these values in the first formula:

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

$latex A=\frac{1}{2}\times 16\times 18$

$latex A=144$

The area is 144 cm².

Start now: Explore our additional Mathematics resources

**EXAMPLE 3**

The area of a scalene triangle is 84 m². If its base is 14 m, what is the length of its height?

##### Solution

We observe the following information:

- Area, $latex A=84$ m²
- Base, $latex b=14$ m

In this case, we want to find the height. Therefore, we use the first formula with these values and solve for *h*:

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

$latex 200=\frac{1}{2}\times (14)\times h$

$latex 84=7 h$

$latex h=12$

The length of the height is 12 m.

**EXAMPLE 4**

A scalene triangle has sides of length 10 m, 12 m, and 14 m. Find the area.

##### Solution

In this case, we have the lengths of the three sides of the triangle:

- Side 1, $latex a=10$ m
- Side 2, $latex b=12$ m
- Side 3, $latex c=14$ m

We use Heron’s formula to find the area. For this, we start by finding the semi perimeter:

$latex S=\frac{a+b+c}{2}$

$latex S=\frac{10+12+14}{2}$

$latex S=\frac{36}{2}$

$latex S=18$

Therefore, we have:

$latex A=\sqrt{S(S-a)(S-b)(S-c)}$

$$A=\sqrt{18(18-10)(18-12)(18-14)}$$

$latex A=\sqrt{18(8)(6)(4)}$

$latex A=\sqrt{18(8)(6)(4)}$

$latex A=\sqrt{3456}$

$latex A=58.8$

The area is 58.8 m².

## Area of a scalene triangle – Practice problems

Put what you have learned into practice and use the formulas for the area of a scalene triangle to solve the following problems. If you need help, you can look at the examples outlined above.

## See also

Interested in learning more about scalene triangles? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**