Review the formulas for volume and apply them to various problems.
Review of Volume Formulas
- Polygonal prisms and cylinders: base area × height
- Polygonal pyramids and cones: (1/3) × base area × height
- Spheres: (4/3)πr³
📘 Example ①
There are a cylinder and a cone with the same base area and height. If the volume of the cylinder is 90π cm³, find the volume of the cone.
Answer: Cone = cylinder × (1/3) = 90π × (1/3) = 30π cm³
There are a cylinder and a cone with the same base area and height. If the volume of the cylinder is 90π cm³, find the volume of the cone.
Answer: Cone = cylinder × (1/3) = 90π × (1/3) = 30π cm³
💡 Key Point
- With the same base area and height, a pyramid is 1/3 of a prism
- Do not confuse the volume and surface area of a sphere (volume uses r³; surface area uses r²)
Practice Problems
- Find the volume of a triangular pyramid with a base area of 12 cm² and a height of 9 cm.
- Find the volume and surface area of a sphere with a radius of 3 cm (leave π as it is).
Answer and Explanation
- Answer: 36 cm³. (1/3)×12×9=36.
- Answer: Volume=36π cm³, surface area=36π cm². Volume=(4/3)π×27=36π. Surface area=4π×9=36π.