Volume of pyramids practice

Sure! Here are two practice problems for finding the volume of pyramids:

Problem 1:
A square-based pyramid has a base length of 8 cm and a height of 12 cm. Find its volume.

Solution:
The volume of a pyramid can be found using the formula V = (1/3) * base area * height.
Since the base of the pyramid is square, the base area is equal to the side length squared.
Therefore, the base area is 8 cm * 8 cm = 64 cm^2.
Substituting the values into the formula, we get V = (1/3) * 64 cm^2 * 12 cm = 256 cm^3.
The volume of the pyramid is 256 cubic centimeters.

Problem 2:
A triangular-based pyramid has a base with sides measuring 5 cm, 6 cm, and 7 cm. The height of the pyramid is 9 cm. Find its volume.

Solution:
The volume of a pyramid can be found using the formula V = (1/3) * base area * height.
Since the base of the pyramid is a triangle, we need to find its area using Heron's formula.
Heron's formula states that the area of a triangle with sides a, b, and c is given by the square root of s(s-a)(s-b)(s-c), where s is the semiperimeter of the triangle.

In this case, a = 5 cm, b = 6 cm, and c = 7 cm.
The semiperimeter is given by s = (a + b + c) / 2 = (5 cm + 6 cm +7 cm) / 2 = 9 cm.

Using Heron's formula, the area of the base triangle is A = √(9 cm * (9 cm - 5 cm) * (9 cm - 6 cm) * (9 cm - 7 cm)).
Calculating this, we get A = √(9 cm * 4 cm * 3 cm * 2 cm) = √(216 cm^2) ≈ 14.6969 cm^2.

Substituting the values into the formula, we get V = (1/3) * 14.6969 cm^2 * 9 cm ≈ 44.0908 cm^3.
The volume of the pyramid is approximately 44.0908 cubic centimeters.

These are the answers for the 7th grade practice and quick check:

1. 48
2. 682.67
3. 240
4. 15
5. 2,480

Quick Checks Answers:
1. 336
2. 480
3. 308, 025
4. 8,000
5. 6.93

THANK YOU SO MUCH!