How to Find the Slant Height of Square Pyramids

How to Find the Slant Height of Square Pyramids
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A square pyramid's ​slant height​ is the distance between its top, or ​apex​, to the ground along one of its sides. You can solve for slant height by visualizing it as one element of a triangle. Doing so, you can use the Pythagorean Theorem to compare slant height to the pyramid's height and side lengths

Finding Slant Height as a Triangle

To solve for slant height, you can understand slant height as one line in a right triangle inside the pyramid. The triangle's other two lines will be the height from the center of the pyramid to its apex, and a line half the length of one of the pyramid's sides that connects the center to the bottom of the slant. The slant length is the side of the triangle opposite to the right angle -- this side is called the ​hypotenuse​.

The ​Pythagorean Theorem​ is a mathematical formula that tells you how the different sides of a right triangle relate to each other. If ​a​ and ​b​ are the two sides connected by the right angle, and ​c​ is the hypotenuse, then:

a^2 + b^2 = c^2

The "2" in the formula signified that you are ​squaring​ the numbers. To square a number means you are multiplying it by itself. So ​c​​2​ is the same as ​c​ × ​c​.

Finding the Height and Base

If you know the height of a pyramid and the length of one of side of its square base, you can use the Pythagorean Theorem to solve for slant height. The "​a​" and "​b​" in the Theorem will be height and half the length of one side, and "​c​" will be slant height, since slant height is the hypotenuse of the triangle:

\text{height}^2 + \text{half length}^2 = \text{slant height}^2

Say you have a pyramid that is 4 inches high, and has a square base with sides 6 inches long. To find half the side length, divide the side length by 2. So this pyramid will have a height of 4 inches and a half length of 3 inches.

Squaring the Height and Base

In the Pythagorean Theorem, the hypotenuse squared is equal to the sum of the squares of the other two sides. Now square the height and the half length, and add the squared numbers together.

Take the pyramid with 4 inch height and 3 inch half length. Square 4 and 3. Remember that a number squared is that number times itself. So:

4^2 + 3^2 = \text{slant height}^2 \\ (4 × 4) + (3 × 3) = \text{slant height}^2

You then add these two numbers together:

16 + 9 = \text{slant height}^2 \\ 25 = \text{slant height}^2

So the slant height squared is equal to 25.

Taking the Square Root

You now know that the slant height squared – or multiplied by itself – is 25. To find the slant height, find the number that, multiplied by itself, equals 25. This is called taking the ​square root​ of 25. If you check small numbers multiplied by themselves, you will find that 5 times 5 is equal to 25. So:

\sqrt{25} = 5 \text{ inches} =\text{ slant height}

It's not always possible to find the square roots of numbers by guessing and checking. Many numbers do not have exact square roots, so you may need a calculator to find an approximation.

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