SURFACE AREAS AND VOLUMES
Frequently Asked Questions
The surface area of a solid is the total area covered by all its outer faces. It represents the amount of material required to cover the solid from the outside.
Volume is the measure of space occupied by a solid object. It indicates the capacity of the solid to hold material such as liquid or gas.
Curved surface area is the area of only the curved part of a solid, excluding any flat circular or polygonal faces.
Total surface area is the sum of the curved surface area and the areas of all flat faces of a solid.
The chapter includes cube, cuboid, right circular cylinder, right circular cone, sphere, hemisphere, hollow solids, and combinations of these solids.
The total surface area of a cube is given by \(6a^2\), where \(a\) is the length of one edge.
The volume of a cuboid is calculated using the formula \(l \times b \times h\), where \(l\), \(b\), and \(h\) are length, breadth, and height respectively.
The curved surface area of a cylinder is \(2\pi rh\), where \(r\) is the radius and \(h\) is the height.
The total surface area of a cylinder is \(2\pi r(h + r)\), which includes the curved surface and both circular ends.
Slant height is the distance from the centre of the base of a cone to a point on the curved surface along the side. It is denoted by \(l\).
Slant height is calculated using \(l = \sqrt{r^2 + h^2}\), where \(r\) is radius and \(h\) is height of the cone.
The volume of a cone is \(\frac{1}{3}\pi r^2 h\).
The surface area of a sphere is \(4\pi r^2\), where \(r\) is the radius.
The volume of a sphere is \(\frac{4}{3}\pi r^3\).
A hemisphere is exactly half of a sphere, having one flat circular face and one curved surface.
The curved surface area of a hemisphere is \(2\pi r^2\).
The total surface area of a hemisphere is \(3\pi r^2\), including the circular base.
Combination of solids refers to objects formed by joining two or more basic solids such as cone and cylinder or sphere and cylinder.
The total volume is obtained by adding or subtracting the volumes of the individual solids depending on the structure.
These problems involve melting or reshaping a solid into another solid without loss of material, so volume remains constant.
The principle of conservation of volume is used, which states that volume before and after transformation remains the same.
Hollow solids have thickness and empty space inside, requiring subtraction of inner volume or surface area from the outer one.
A hollow cylinder has an outer radius, inner radius, and thickness, commonly used in pipes and tubes.
Volume is calculated as \(\pi h(R^2 - r^2)\), where \(R\) is outer radius and \(r\) is inner radius.
All dimensions must be in the same unit to avoid incorrect results in surface area or volume calculations.
Applications include water tanks, packaging boxes, ice-cream cones, pipes, spherical balls, containers, and construction materials.
Such problems require calculation of surface area since only the outer surface is coated.
Capacity problems involve volume, while surface area problems involve covering or coating material.
Formula-based numericals, word problems, combination of solids, recasting solids, and application-based questions are common.
By clearly identifying the solid, choosing the correct formula, maintaining unit consistency, and writing steps systematically.
The formulas are fixed and questions are predictable, making it easier to score high with proper practice.
Students should revise formulas, practice mixed numericals, and focus on real-life application problems.
It enhances spatial reasoning, numerical accuracy, logical thinking, and real-world problem interpretation.
It directly applies geometry to real objects and measurements encountered in everyday situations.
p is used because many solids like cylinders, cones, and spheres involve circular bases or curved surfaces.
