Revision: Mensuration >> Surface Areas and Volumes Maths English Medium Class 9 CBSE

• Cube: A cube is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meetings at each vertex. The cube is the only regular hexahedron (i.e., a solid figure with six plane faces) and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.
• Lateral surface area of the cube: Out of the six faces of a cube, we only find the area of the four faces, leaving the bottom and top faces. In such a case, the area of these four faces is called the lateral surface area of the cube.

The solid obtained on revolving a right-angled triangle about one of its sides (other than the hypotenuse) is called a cone or a right circular cone.

A sphere is a solid obtained by revolving a circle about any one of its diameters.
The radius of the sphere is equal to the radius of the circle revolved.

A cylinder is a three-dimensional solid figure that has two identical circular bases joined by a curved surface at a  particular distance from its centre, which is its height.

The solid obtained on revolving a right-angled triangle about one of its sides (other than the hypotenuse) is called a cone or a right circular cone.

A sphere is a solid obtained by revolving a circle about any one of its diameters.
The radius of the sphere is equal to the radius of the circle revolved.

• Cuboid: A cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. A cuboid looks like a rectangular box. It has 6 faces. Each face has 4 edges. Each face has 4 corners (called vertices).
• Surface of a cuboid: the outer surface of a cuboid is made up of six rectangles (in fact, rectangular regions, called the faces of the cuboid), whose areas can be found by multiplying the length by breadth for each of them separately and then adding the six areas together.
• Lateral surface area of the cuboid: Out of the six faces of a cuboid, we only find the area of the four faces, leaving the bottom and top faces. In such a case, the area of these four faces is called the lateral surface area of the cuboid.

• Total surface area of cuboid = 2(lb + bh + lh)
• The lateral surface area of a cuboid = 2h(l + b)

• Total surface area of the cube = 6a².
• Lateral surface area of a cube = 4a².

• Volume = \[\frac{1}{3}\pi r^2h\]

• Curved (lateral) surface area = πrl or \[\pi r\sqrt{h^{2}+r^{2}}\]
  (\[l=\sqrt{h^2+r^2}\])
  

• Total surface area = $\pi r(l+r)$

Sphere (radius = r)

• Volume =\[\frac{4}{3}\pi r^3\]

• Surface Area = 4πr²

Spherical Shell (external radius R, internal radius r)

• Thickness = R − r

• Volume of material = \[\frac{4}{3}\pi(R^3-r^3)\]

• Volume of a Cuboid = l × b × h

Curved surface area of a cylinder = circumference of base × height

                                                        = 2πrh

Total surface area = Curved surface area + 2 (Area of cross-section)

                               = 2πrh + 2πr²

                               = 2πr(r + h)

Volume = Area of cross-section × height (or, length)

              = πr²h

• Volume = \[\frac{1}{3}\pi r^2h\]

• Curved (lateral) surface area = πrl or \[\pi r\sqrt{h^{2}+r^{2}}\]
  (\[l=\sqrt{h^2+r^2}\])
  

• Total surface area = $\pi r(l+r)$

Sphere (radius = r)

• Volume =\[\frac{4}{3}\pi r^3\]

• Surface Area = 4πr²

Spherical Shell (external radius R, internal radius r)

• Thickness = R − r

• Volume of material = \[\frac{4}{3}\pi(R^3-r^3)\]