Video: Finding the Height of a Pyramid

Determine, to the nearest hundredth, the height of the solid formed by the given net.

03:41

Video Transcript

Determine, to the nearest hundredth, the height of the solid formed by the given net.

Let’s begin by having a look at the net. The shape in the center of our diagram has three lines that have a double dash through them. This means that these three sides are of equal length. We can also say that there are two other lines in our net that are the same length. So when we’re folding our net together, the top edge of our top triangle will fold to the top edge of the shape on the bottom. And the bottom edge on the triangle will fold into the bottom edge of the shape. So the shape at the bottom of our net must be a square, as we have four equal sides. And this three-dimensional shape that’s made from the net must be a square-based pyramid. It will have a square on the bottom and four isosceles triangles.

In this question, we need to find the height of the solid so we could draw that on our diagram. And we can note that the height of the triangle formed inside our pyramid must have a right angle. Since it’s made of the horizontal line on the ground and the vertical line pointing straight upwards. Let’s now take a closer look at this internal triangle. We can use the letter ℎ to represent the value of the height that we want to work out. The base of our triangle can be found by half of the square on the bottom. And since the length of the square is 12, this means that halfway across would be six centimeters. The hypotenuse of our triangle will be the same as the slant height of the pyramid, which is given by the height of one of the original triangles in the pyramid. And that’s 15.26 centimeters.

As we look at our triangle then, we can see it’s a right-angled triangle. We know two of the lengths and we want to find the third length. So we can use the Pythagorean theorem. Which says that the square of the hypotenuse is equal to the sum of the squares on the other two sides. So we can start by writing our formula and then filling in the values. Our longest side, the hypotenuse, is equal to 15.26 squared, which we can say is equal to ℎ squared plus six squared. And it doesn’t matter which way round we have our ℎ and our six. Using our calculator to evaluate the squares will give us 232.8676 equals ℎ squared plus 36.

Next, to find ℎ squared by itself, we subtract 36 from both sides of our equation, giving us 232.8676 minus 36 equals ℎ squared. Evaluating the left-hand side will give us 196.8676 equals ℎ squared. And now since we have ℎ squared, then to find ℎ by itself, we do the inverse operation which is to take the square root of both sides of the equation. Taking the square root of 196.8676 will give us ℎ equals 14.0309515. However, we’re not quite finished as we’ve been asked to round it to the nearest hundredth.

Since our hundredth is our second decimal place, this means that our answer will have two decimal places. So we check our third decimal place. That’s our thousandth column. And if this digit is five or more, then we will round up. In this case, it isn’t since the digit is zero. It’s less than five. So our answer stays as 14.03, with the length unit of centimeters. So our final answer is that the height of the solid formed by the net is 14.03 centimeters.

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