Question

What is the volume of a 4-sided pyramid if its base is \(9 \mathrm{~cm}^2\) and its perpendicular height is \(5 \mathrm{~cm}\) (take \(\pi\) to be \(3.14\) )?

Short answer

The volume of the pyramid is 15 cm³.

Step-by-step solution

Identify the Formula for Volume

The volume of a pyramid is given by the formula \[ V = \frac{1}{3} \times B \times h \]where:- \( B \) is the area of the base- \( h \) is the perpendicular height.

Substitute the Given Values

Insert the given values of the base area and height into the formula:\[ V = \frac{1}{3} \times 9 \, \text{cm}^2 \times 5 \, \text{cm} \]

Calculate the Volume

Perform the multiplication and division to find the volume:\[ V = \frac{1}{3} \times 45 \, \text{cm}^3 \]\[ V = 15 \, \text{cm}^3 \]

Key concepts

pyramid volume formula

To find the volume of a pyramid, you need to use a specific formula that relates the area of the base, the height, and a constant multiplication factor. The formula for the volume of a pyramid is \( V = \frac{1}{3} \times B \times h \).
Here:

• \(B\): The area of the base of the pyramid
• \(h\): The perpendicular height of the pyramid

Notice that the formula includes a factor of \(\frac{1}{3}\). This indicates that the volume of a pyramid is one-third of the volume of a prism that has the same base area and height.
An example: if the base area \( B \) is \(9 \text{cm}^2\) and the height \( h \) is \(5 \text{cm}\), substituting these values into our formula: \( V = \frac{1}{3} \times 9 \times 5 \). This gives \( V = 15 \text{cm}^3\). Calculating the volume using this formula is straightforward once you've identified the base area and height. Teaching this formula and practicing with different values is crucial for understanding.

geometry calculation

Geometry is a branch of mathematics that deals with shapes, sizes, and properties of figures. Calculating the volume of a pyramid is a fundamental exercise in geometric calculations, helping students understand three-dimensional shapes.
When dealing with geometric calculations like the volume of a pyramid, always:

• Identify the shape of the base
• Calculate or find the area of the base
• Determine the height
• Apply the appropriate formula

For a four-sided pyramid, the base is often a square or rectangular shape. Calculating the area of a square base involves squaring the length of one side. For a rectangular base, multiply the lengths of two adjacent sides.
These basic steps are foundational and help highlight how different geometric shapes are related to one another. Performing these calculations strengthens problem-solving skills and provides a deeper understanding of mathematical concepts.

mathematics education

Mathematics education is about more than just solving problems—it's about developing a thorough understanding of underlying principles and concepts. Learning how to calculate the volume of a pyramid is one way to develop these skills.
In teaching this topic, focus on:

• Explaining why the formula \( V = \frac{1}{3} \times B \times h \) works
• Providing various examples with different base shapes and sizes
• Encouraging hands-on activities or visual tools like 3D models

By showing students how and why these formulas work, we deepen their comprehension and make the learning process more engaging. Using visual aids can also help students grasp abstract concepts more concretely.
Encouraging experimentation and practical application helps solidify these ideas. For example, students can measure actual objects and calculate their volumes, bridging the gap between theory and real-world application. This approach promotes a holistic understanding of mathematics and its practical utility.