Heron's Formula
Definition
Heron's formula is a formula that can be used to find the area of a triangle, when given its three side lengths. It can be applied to any shape of triangle, as long as we know its three side lengths. The formula is as follows:
The area of a triangle whose side lengths are and is given by
where , semi-perimeter of the triangle.
Other useful forms are
\[\begin{align} A&=\frac 1 4\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}\\ \\ A&=\frac 1 4\sqrt{ \big[(a+b+c)(a+b-c) \big] \times \Big[\big(+(a-b)+c\big)\big(-(a-b)+c\big) \Big]}\\
A&=\frac 1 4\sqrt{\Big[(a+b)^2-c^2\Big] \times \ \Big[c^2-(a-b)^2\Big] }\\ \\
A&=\frac{1}{4}\sqrt{4a^2b^2-\big(a^2+b^2-c^2\big)^2}\\ A&=\frac{1}{4}\sqrt{2\left(a^2 b^2+a^2c^2+b^2c^2\right)-\left(a^4+b^4+c^4\right)} \\ A&=\frac{1}{4}\sqrt{\left(a^2+b^2+c^2\right)^2-2\left(a^4+b^4+c^4\right)}. \end{align}\]
Although this seems to be a bit tricky (in fact, it is), it might come in handy when we have to find the area of a triangle, and we have no other information other than its three side lengths.
Proof of Heron's Formula
This formula follows from the area formula .
By the law of cosines, .
Substituting into the Pythagorean identity yields Heron's formula (after a series of algebraic manipulations).
Examples
Find the area of the triangle below.
Imgur
Since the three side lengths are all equal to 6, the semiperimeter is . Therefore the area of the triangle is
Find the area of the triangle below.
Imgur
Since the three side lengths are 4, 5, and 7, the semiperimeter is . Therefore the area of the triangle is
What is the area of a triangle with side lengths 13, 14, and 15?
Since the three side lengths are 13, 14, and 15, the semiperimeter is . Therefore the area of the triangle is
Find the area of the triangle below.
Imgur
Since the three side lengths are 6, 8, and 10, the semiperimeter is . Therefore the area of the triangle is
Find the area of a triangle with side lengths and .
We have and . Hence,
Find the area of the triangle outlined in black.
Image
We can use the Pythagorean theorem to find that the side lengths are .
If we used the direct form of , we will quickly get into a huge mess because these lengths are not integers.Instead, we will use an alternate form of Heron's formula:
Note: This triangle appears in Composite Figures, which is an easier approach.
Find the area of the triangle below.
Imgur
Since the three side lengths are all equal to 6, the semiperimeter is . Therefore the area of the triangle is
Find the area of the triangle below.
Imgur
Since the three side lengths are 4, 5, and 7, the semiperimeter is . Therefore the area of the triangle is
What is the area of a triangle with side lengths 13, 14, and 15?
Since the three side lengths are 13, 14, and 15, the semiperimeter is . Therefore the area of the triangle is
Find the area of the triangle below.
Imgur
Since the three side lengths are 6, 8, and 10, the semiperimeter is . Therefore the area of the triangle is
Additional Problems
What is the area of a triangle with sides of length 13, 14, and 15?
If has side lengths 10, 8, and 4, then the area of the triangle can be expressed as , where is a -digit number. Find .
In the figure to the right, the areas of the squares and are 388, 153, and 61, respectively.
Find the area of the blue triangle.