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A right pyramid with a square base has a volume of 252 cubic centimeters. The length of one of the sides of its base is 6 centimeters. Rounded to the nearest centimeter, what is the vertical height of the pyramid?

A right pyramid with a square base has a volume of $252$ cubic centimeters. The length of one of the sides of its base is $6$ centimeters. Rounded to the nearest centimeter, what is the vertical height of the pyramid?

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Pythagorean theorem

Full solution

Q. A right pyramid with a square base has a volume of

252

cubic centimeters. The length of one of the sides of its base is

6

centimeters. Rounded to the nearest centimeter, what is the vertical height of the pyramid?

Understand formula for volume: Understand the formula for the volume of a pyramid. $\newline$ The volume $V$ of a pyramid with a square base is given by the formula $V = \frac{1}{3} \times \text{base area} \times \text{height}$ . $\newline$ Here, the base area is the area of the square base, which can be calculated as side length squared.
Plug in given values: Plug in the given values into the volume formula. $\newline$ We know the volume $V = 252 \, \text{cm}^3$ and the side length of the base is $6 \, \text{cm}$ . $\newline$ First, calculate the base area: base area = side length $*$ side length = \(6 \, \text{cm} $*$ $6$ \, \text{cm} = $36$ \, \text{cm}^ $2$ \).
Use volume formula for height: Use the volume formula to find the height. $\newline$ We have $V = \frac{1}{3} \times \text{base area} \times \text{height}$ , which means $252 \, \text{cm}^3 = \frac{1}{3} \times 36 \, \text{cm}^2 \times \text{height}$ .
Solve for height: Solve for the height. $\newline$ Multiply both sides of the equation by $3$ to get rid of the fraction: $3 \times 252 \, \text{cm}^3 = 36 \, \text{cm}^2 \times \text{height}$ . $\newline$ This simplifies to $756 \, \text{cm}^3 = 36 \, \text{cm}^2 \times \text{height}$ .
Divide to isolate height: Divide both sides by the base area to isolate the height. $\newline$ $756 \, \text{cm}^3 / 36 \, \text{cm}^2 = \text{height}$ . $\newline$ This simplifies to $\text{height} = 21 \, \text{cm}$ .
Round to nearest centimeter: Round the height to the nearest centimeter. $\newline$ The height is already an integer value, so rounding to the nearest centimeter does not change the value: $\text{height} \approx 21\,\text{cm}$ .

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