Question Video: Finding the Area of a Triangle Using Heron’s Formula Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle, where 𝐴𝐶 is equal to seven centimeters, 𝐵𝐶 plus 𝐴𝐵 is equal to 28 centimeters, and 𝐵𝐶 minus 𝐴𝐵 is equal to four centimeters. Find the area of 𝐴𝐵𝐶, giving your answer to the nearest square centimeter.

Let’s begin by sketching the triangle. We’re given that the side length 𝐴𝐶 is seven centimeters. And to find the area, we’ll first need to find the side lengths 𝐴𝐵 and 𝐵𝐶. We can then use these in Heron’s formula to find the area. If we call our sides lowercase 𝑎, 𝑏, and 𝑐, then from the information in the question, we have that 𝑏 is equal to seven centimeters, that’s the side 𝐴𝐶; we have 𝑎 plus 𝑐 is 28 centimeters, that is, side lengths 𝐵𝐶 and 𝐴𝐵 sum to 28; and that 𝑎 minus 𝑐 is four centimeters, that is, the side length 𝐵𝐶 minus the side length 𝐴𝐵 is four centimeters.

Let’s call these equations one and two. And we can solve these for 𝑎 and 𝑐. If we begin with equation two and add 𝑐 to both sides, we’re left with 𝑎 is equal to four plus 𝑐. So we have 𝑎 in terms of 𝑐. Now substituting this into equation one, we have four plus 𝑐 plus 𝑐 is equal to 28. And collecting our 𝑐’s together, we have four plus two 𝑐 is 28. Subtracting four from both sides, that’s two 𝑐 is 28 minus four, which is 24. And now dividing both sides by two, that gives us 𝑐 is equal to 12. That is, lowercase 𝑐, which is our side length 𝐴𝐵, is 12 centimeters.

So now making some space and making a note of this, we can substitute 𝑐 equal to 12 into either of our equations to find the value of 𝑎. So let’s use equation one. And with 𝑐 as 12, we have 𝑎 plus 12 is 28. Subtracting 12 from both sides, we have then 𝑎 is equal to 28 minus 12, which gives us 𝑎 is equal to 16 centimeters. And remember this is the side length 𝐵𝐶.

So now we have our three side lengths. 𝑎 is 16 centimeters, 𝑏 is seven centimeters, and 𝑐 is 12 centimeters. And we want to use these to find the area of our triangle. To do this, we’re going to use Heron’s formula. This tells us that for a triangle with side lengths lowercase 𝑎, 𝑏, and 𝑐, the area of that triangle is the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is the semiperimeter of the triangle, that is, half the perimeter. And remember that the perimeter of a triangle, or indeed any polygon, is the sum of its side lengths. In our case then, the semiperimeter 𝑠 is equal to 16 plus seven plus 12 all divided by two, that is, 35 over two, which is 17.5.

So now we have everything we need to use Heron’s formula. This gives us that our area is equal to the square root of 17.5, which is 𝑠, multiplied by 17.5 minus 16, which is 𝑠 minus 𝑎, multiplied by 17.5 minus seven, that’s 𝑠 minus 𝑏, multiplied by 17.5 minus 12, that’s 𝑠 minus 𝑐. And evaluating each of our parentheses, this gives us the square root of 17.5 multiplied by 1.5 multiplied by 10.5 multiplied by 5.5. The argument of our square root evaluates to 1515.9375.

And taking the positive square root since areas are always positive, to four decimal places, the area of our triangle 𝐴𝐵𝐶 is 38.9350 centimeters squared. Hence, to the nearest square centimeter, our triangle’s area is 39 centimeters squared.