Find the area of the figure below using Heron’s formula, giving the answer to three decimal places.
So if we look at the question, it actually asks us to use Heron’s formula. But first of all, we need to see what is Heron’s formula. Well, Heron’s formula is actually a way of finding out the area of a triangle. And it’s particularly useful when we don’t know the perpendicular height of a triangle and also we haven’t got any angles involved with that triangle.
So if we have a triangle 𝑎, 𝑏, and 𝑐, then Heron’s formula tells us that the area of a triangle is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐. And this is where 𝑠 is actually our semiperimeter. So what this means is half the distance around the outside of the triangle.
And to actually calculate that, what we do is we say that 𝑠 is equal to 𝑎 plus 𝑏 plus 𝑐. So that’s all the sides added together then divided by two. Okay, great, so now we have Heron’s formula. And we know about the semiperimeter. Let’s use these to find the area of our shape.
So the first thing I’ve done is actually divided our shape into two parts. So we have part 𝑎 and part 𝑏 cause that it’s actually made up of two triangles. And before we actually find the area of either of these triangles, what we’re gonna need to do is actually find this length here, which I’ve called 𝑥.
Well, to actually find 𝑥, what we can say is that this triangle here is actually a right triangle. And that’s because we have a right angle. So therefore, what we can actually do is use the Pythagorean theorem.
And what the Pythagorean theorem tells us is that if we have a right triangle with sides 𝑎, 𝑏, 𝑐, where 𝑐 is actually the hypotenuse, so the longest side opposite the right angle, then what we’re gonna get is that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. So the squares of the two shorter sides added together is equal to the square of our hypotenuse.
Okay, so like I said, let’s use this to find 𝑥. So what I’ve done here is I’ve actually got 𝑥 squared is equal to 20 squared minus 16 squared. So as you can see, I’ve actually rearranged the Pythagorean theorem because we’ve actually trying to find here a shorter side.
So therefore, it’s gonna to be the longest side squared minus one of the other shorter sides squared. And we know that in this case, the 20 is going to be our longest side or our hypotenuse because it’s actually opposite the right angle.
Okay, great, so now, let’s find 𝑥. So we’re gonna get 𝑥 squared is equal to 144. So now, if we actually take the square root of each side, we’re gonna get 𝑥 is equal to 12. So great, we’ve now found our missing side. So now, what I need to do is move on and actually find the area of triangle 𝑎 and the area of triangle 𝑏.
So we’re gonna start with triangle 𝑎. It’s worth looking at the question here again to remind ourselves. Well, the question says that we need to use Heron’s formula. So therefore, it doesn’t matter if there’s another method we can use. We do need to do use this method to actually find the area of our shape. So the first thing we need to do is actually find out the semiperimeter.
So we can say that the semiperimeter is gonna be equal to the three sides added together, so 12 plus 16 plus 20 then all divided by two, which is gonna be equal to 24. Okay, so now, what we can do is actually use this in Heron’s formula to find the area of triangle 𝑎.
So we can say that the area is equal to the square root of 24 multiplied by 24 minus 12 multiplied by 24 minus 16 multiplied by 24 minus 20, which is equal to the square root of 9216, which gives us an area of 96 centimeters squared. Okay, so great, we found the area of triangle 𝑎. We can now move on to triangle 𝑏.
So for triangle 𝑏, we’d actually do the same things we did with triangle 𝑎, which is first of all to find the semiperimeter. So this is equal to 16 plus 20 plus 23 all over two which is equal to 59 over 2, which gives us a value of 𝑠 of 29.5.
Okay, great, so now, again, we can actually apply Heron’s formula to find the area of triangle 𝑏. So we’re gonna get that the area is equal to the square root of 29.5 multiplied by 29.5 minus 16 multiplied by 29.5 minus 20 multiplied by 29.5 minus 23, which is gonna give us an answer of 156.818167 centimeter squared for the area of triangle 𝑏.
Okay, fab, so we’ve actually found the area of triangle 𝑎 and triangle 𝑏. So now, what we can actually do is move on to the final stage — that’s to find the actual total area of our shape. Well, to find the actual total area of our shape, what we’re gonna have is 96 plus 156.818167, which is gonna give us 252.818167.
However, have we finished there? Well, no, because if we actually check back to the question, the question says leave your answer to three decimal places. So therefore, we can say that the area of the figure is actually gonna be 252.818 centimeter squared. And that’s to three decimal places.
And we actually found that because if you look at our original answer, we get to the third decimal place, which is the eight. Then, the number after that is a one. And because this is less than five, then the eight will stay the same.
So we can say that our final area is 252.818 centimeters squared to three decimal places.