Lesson Explainer: Heron’s Formula | Nagwa Lesson Explainer: Heron’s Formula | Nagwa

# Lesson Explainer: Heronβs Formula Mathematics • Second Year of Secondary School

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In this explainer, we will learn how to use Heronβs formula to find the area of a triangle.

Before examining Heronβs formula in detail, letβs look at two other formulas we can use to find the area, , of a triangle. The formula we choose will depend on the information that we have about the triangle.

Recall that, to calculate the area of any triangle that has a known base length and perpendicular height, we can use the formula where is the base length and is the perpendicular height. To find the area of the triangle below, we can substitute 12 cm into this formula for and 4 cm for :

Also remember that is another formula we can use to calculate a triangleβs area. In it, and are the lengths of two of the triangleβs sides, and is the measure of the included angle, or the angle formed by the sides with lengths and .

We can use this formula to find the area of any triangle that has an angle with a known measure and that is formed by two sides with known lengths. To find the area of the following triangle, we can substitute 15 cm and 18 cm into the formula for and and for :

Now, suppose that we know the lengths of all three sides of a triangle but the measures of none of its angles. With only this information, we can use neither the formula nor the formula to calculate the triangleβs area. However, we can use Heronβs formula.

### Definition: Heronβs Formula

Heronβs formula states that the area, , of a triangle with side lengths of , , and is where is the semiperimeter of the triangle, or half its perimeter. The triangleβs semiperimeter is given by the formula

Letβs test Heronβs formula to make sure that we get the same area, , for an equilateral triangle with a side length of 2 cm that we get when using the formula

In this formula, the variable is the triangleβs side length. Substituting into this formula gives us

We get cm2 for the triangleβs area. To make our comparison, we will now use Heronβs formula to find the area. To do this, first, we will need to calculate the semiperimeter:

Heronβs formula then gives us

We see that the triangleβs area is cm2, the same answer that we arrived at previously. This is what we would expect.

In the examples that follow, we will look at some other instances in which Heronβs formula is used to calculate not only the areas of triangles but also the areas of figures composed of triangles.

### Example 1: Finding the Area of a Triangle Using Heronβs Formula

The area of the triangle whose side lengths are 3 cm, 6 cm, and 7 cm equals cm2.

In this problem, we are given only the three side lengths of a triangle. Since we are given neither the triangleβs height nor any of its angle measures, we must use Heronβs formula to find its area. Recall that Heronβs formula states that the area, , of a triangle with side lengths of , , and is where is the triangleβs semiperimeter, or half its perimeter. The semiperimeter can be found using the formula

Letβs begin by finding the triangleβs semiperimeter. We must substitute each of the triangleβs side lengths into the semiperimeter formula for either , , or . It does not matter which side length we substitute for which variable. Here, we will substitute , , and and then simplify to get a semiperimeter of

Now, we can substitute values into Heronβs formula to find the triangleβs area.

Since , , , and , Heronβs formula gives us

Thus, we can say that the area of the triangle whose side lengths are 3 cm, 6 cm, and 7 cm is cm2.

In the next example, we will also calculate the area of a triangle with three given side lengths using Heronβs formula.

### Example 2: Finding the Area of a Triangle Using Heronβs Formula

is a triangle, where , , and . Find the area of giving the answer to the nearest square centimetre.

First, letβs draw a sketch of triangle . We are told that , , and .

Since we know the triangleβs three side lengths, we can use Heronβs formula to find its area. According to Heronβs formula, the area, , of a triangle with side lengths of , , and is where is the triangleβs semiperimeter. The semiperimeter can be calculated using the formula

Substituting , , and into the semiperimeter formula gives us

Now, since , , , and , Heronβs formula tells us that the triangleβs area is

The area of triangle to the nearest square centimetre is 235 cm2.

Next, we will use Heronβs formula to calculate the area of a rhombus. Remember that the sides of a rhombus all have the same length.

### Example 3: Finding the Area of a Rhombus Using Heronβs Formula

The perimeter of the given rhombus is 292 cm and the length of is 116 cm. Use Heronβs formula to calculate the area of the rhombus, giving the answer to three decimal places.

Recall that Heronβs formula is where is the area of a triangle that has sides with lengths of , , and and a semiperimeter of . The semiperimeter is given by the formula

In order to use Heronβs formula to calculate the area of the given rhombus, we must decompose the rhombus into two triangles: triangle and triangle . Since a rhombus is a quadrilateral that has four sides of equal length, each of the sides of rhombus must have a length of

Therefore, we know that the lengths of the sides of triangle are

With this information, we can use Heronβs formula to calculate the triangleβs area. We can begin by finding its semiperimeter:

Next, we can substitute , , , and into Heronβs formula to get an area of

We know that triangle must have the same area as triangle , because the two triangles have the same side lengths. Therefore, since rhombus is composed of the two triangles, the rhombusβs area must be equal to

The area of the rhombus to three decimal places is 5βββ142.085 cm2.

In the example that follows, we will calculate the area of a figure composed of two triangles using Heronβs formula. One of the two triangles is a right triangle, and we will need to use the Pythagorean theorem to determine one of its leg lengths.

### Example 4: Finding the Area of a Quadrilateral Using Heronβs Formula

Find the area of the figure below using Heronβs formula, giving the answer to three decimal places.

We know that Heronβs formula is where is the area of a triangle that has sides with lengths of , , and and a semiperimeter of . We also know that the semiperimeter can be calculated with the formula

In order to use Heronβs formula to calculate the area of the figure, we must decompose it into a pair of triangles.

One of the triangles into which the figure can be decomposed has side lengths of 16 cm, 20 cm, and 23 cm. Letβs begin by finding the area of this triangle using Heronβs formula. Substituting the triangleβs three side lengths into the semiperimeter formula gives us

Next, substituting the triangleβs three side lengths and its semiperimeter into Heronβs formula gives us

Looking at the other triangle, we can see that it is a right triangle with a leg that is 16 cm long and a hypotenuse that is 20 cm long. However, we are not given the length of the other leg of the triangle.

In order to find the length, we can use the Pythagorean theorem, which states that if a right triangle has legs of lengths and and a hypotenuse of length , then

We can substitute and , and then solve for to determine the length as follows:

Note that we only need to consider the positive root of 144, since a length cannot be negative. Thus, the lengths of the three sides of the second triangle are

Now that we know all three side lengths, letβs find the triangleβs area. First, we can substitute the side lengths into the semiperimeter formula to get

Next, we can substitute the side lengths and the semiperimeter into Heronβs formula to get

Note that we could have also used the formula to find the triangleβs area, since it is a right triangle with a base of 12 cm and a height of 16 cm. This formula also gives us an area of

Combining the areas of both triangles gives us a total area of .

The area of the figure to three decimal places is 252.818 cm2.

In the final example, we use Heronβs formula to help us find the radius of a circle inscribed in a triangle.

### Example 5: Using Heronβs Formula to Find the Radius of a Circle That is Inside a Triangle

The lengths of a triangle are 12 cm, 5 cm, and 11 cm. Find the radius of the interior circle touching the sides using the formula , where is half of the triangleβs perimeter.

According to the formula in order to calculate the radius of the interior circle touching the sides of the triangle described in the problem, we need to know both the triangleβs area and half its perimeter.

Since we are given the side lengths of the triangle, we can use Heronβs formula to calculate its area. Recall that Heronβs formula states that the area, , of a triangle with side lengths of , , and and a semiperimeter of is

The semiperimeter can be found using the formula

Letβs begin by finding the triangleβs semiperimeter. Note that this will also be the value of in because it is half the triangleβs perimeter. Using the triangleβs side lengths of 12 cm, 5 cm, and 11 cm, its semiperimeter is

Now, we can substitute the triangleβs side lengths into Heronβs formula for , , and , and its semiperimeter for , giving

Since we have now established that half the perimeter of the triangle described in the problem is 14 cm and that its area is cm2, we can use the formula to calculate the radius of the interior circle touching the triangleβs sides. Substituting into the formula gives us

The radius of the circle is cm.

Now letβs finish by recapping some key points.

### Key Points

• Heronβs formula states that the area, , of a triangle with side lengths of , , and is where is the semiperimeter of the triangle, or half its perimeter.
• A triangleβs semiperimeter is given by the formula where , , and are its side lengths.
• The radius of the interior circle touching the sides of a triangle can be calculated using the formula where is half of the triangleβs perimeter.

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