Question

The base of a pyramid is a triangle with a base of 12 inches and a height of 8 inches. The height of the pyramid is 10 inches. Find the volume of the pyramid.

Short answer

The volume of the pyramid is 160 cubic inches.

Step-by-step solution

Understand the Formula for Volume of a Pyramid

The volume of a pyramid can be calculated using the formula: \( V = \frac{1}{3} Bh \), where \( B \) is the area of the base and \( h \) is the height of the pyramid.

Calculate the Area of the Base

The base of the pyramid is a triangle. The area of a triangle is calculated by the formula: \( A = \frac{1}{2} \times \, \text{base} \times \, \text{height} \). Substituting the given values, we have \( A = \frac{1}{2} \times 12 \times 8 = 48 \, \text{square inches} \).

Calculate the Volume of the Pyramid

With the area of the base calculated as 48 square inches and the height of the pyramid given as 10 inches, substitute into the formula for the volume: \( V = \frac{1}{3} \times 48 \times 10 \). This simplifies to \( V = 160 \, \text{cubic inches} \).

Key concepts

Understanding a Triangular Base

A pyramid's base can come in various shapes, but in this exercise, we are focusing on a triangular base. A triangle is a three-sided polygon, and it's classified based on the lengths of its sides or the measures of its angles. In mathematics, when calculating areas and volumes, we often encounter triangles as they form a basic geometric shape.

For our specific problem, the triangle serving as the base of the pyramid has a given base length of 12 inches and a height of 8 inches.

When solving problems involving triangular bases, you often have to apply the correct formulas to understand how to calculate area or use them as part of larger formulas for volumes. Geometry basics tell us that the triangle's base and height are perpendicular to each other. This understanding is crucial when it comes to working out the area formula for triangles.

Performing an Area Calculation

Calculating the area of a triangle is essential when dealing with problems involving pyramids with triangular bases. The area (A) of a triangle can be found using the formula: \( A = \frac{1}{2} \times \text{base} \times \text{height} \).

In this example, you multiply half of the base of the triangle by its height. Specifically, we calculate the area of our triangle as: \( A = \frac{1}{2} \times 12 \times 8 = 48 \text{ square inches} \).

This means the triangle that forms the base of the pyramid has an area of 48 square inches. It's essential to correctly calculate this area as it directly affects the computation for the pyramid's volume. Take care to ensure you correctly apply the formula and substitute the right values.

Applying the Volume Formula

Once we have the triangle's area, we can use this to calculate the pyramid's volume. The formula for the volume (V) of a pyramid is: \( V = \frac{1}{3} Bh \), where \( B \) is the area of the base and \( h \) is the height of the pyramid.

Using our calculated area for the base (48 \text{ square inches}) and the given pyramid height (10 inches), we can substitute into the volume formula:\( V = \frac{1}{3} \times 48 \times 10 \).

This results in a volume of \( V = 160 \text{ cubic inches} \).

This step shows how the area of the base integrates into the volume calculation. Understanding the geometry involved and how each part of the formula relates to the physical dimensions of the pyramid is crucial for deriving an accurate measurement. Using precise calculations, students can confidently determine that the volume of this pyramid is 160 cubic inches.