Understanding the Formulas for Volume of Pyramids and Cones
The volume of a pyramid is given by the formula: V = (1⁄3) * base area * height. This formula applies to all types of pyramids, including triangular and square-based pyramids.
On the other hand, the volume of a cone is given by the formula: V = (1⁄3) * π * radius^2 * height. This formula is used to calculate the volume of cones, which are essentially three-dimensional shapes with a circular base and a single vertex.
In this worksheet, we will be applying these formulas to solve problems involving the volume of pyramids and cones.
Solutions
- Find the volume of a square-based pyramid with a base side length of 5 cm and a height of 8 cm.
📝 Note: First, we need to find the base area of the pyramid. Since the base is a square, the area is simply the side length squared.
Base area = 5 cm * 5 cm = 25 cm^2
Now we can use the formula to find the volume:
V = (1⁄3) * 25 cm^2 * 8 cm = (1⁄3) * 200 cm^3 = 66.67 cm^3
- Find the volume of a cone with a radius of 3 cm and a height of 6 cm.
📝 Note: First, we need to square the radius to find the value of radius^2.
Radius^2 = 3 cm * 3 cm = 9 cm^2
Now we can use the formula to find the volume:
V = (1⁄3) * π * 9 cm^2 * 6 cm = (1⁄3) * 3.14 * 54 cm^3 = 57.12 cm^3
- Find the volume of a triangular-based pyramid with a base side length of 6 cm and a height of 9 cm.
📝 Note: First, we need to find the base area of the pyramid. Since the base is a triangle, the area is simply half the product of the base and height of the triangle.
Assuming the triangle is isosceles with equal sides of 6 cm, we can find the height of the triangle using the Pythagorean theorem:
Height of triangle = √(6^2 - 3^2) = √(36 - 9) = √27 = 5.19 cm
Now we can find the base area:
Base area = (1⁄2) * 6 cm * 5.19 cm = 15.57 cm^2
Finally, we can use the formula to find the volume:
V = (1⁄3) * 15.57 cm^2 * 9 cm = (1⁄3) * 140.13 cm^3 = 46.71 cm^3
Summary
In this worksheet, we applied the formulas for the volume of pyramids and cones to solve problems involving these shapes. We also learned how to find the base area of triangles and squares, which is necessary for using the volume formulas. By following these steps and applying the formulas correctly, we can solve a wide range of problems involving the volume of pyramids and cones.
What is the formula for the volume of a pyramid?
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V = (1⁄3) * base area * height
What is the formula for the volume of a cone?
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V = (1⁄3) * π * radius^2 * height
How do I find the base area of a triangle?
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Base area = (1⁄2) * base * height of triangle
Related Terms:
- Volume of Pyramids Worksheet pdf
