Mensuration is an area of mathematics that deals with the analysis of various geometrical shapes. It also discusses the areas and volumes of such geometrical figures.
In this section, let us learn about the surface area of three-dimensional objects like the right circular cylinder, hollow cylinder, right circular cone, etc.,
Right circular cylinder:
A cylinder whose bases are circular in shape and the axis joining the two centres of the bases perpendicular to the planes of the two bases is called a right circular cylinder.
- Volume of a right circular cylinder \(=\) \(πr^2h\) cu. units.
Where \(r\) is the radius and \(h\) is the height.
Hollow cylinder:
A cylinder emptied from the inner side and has a difference in the outer and inner radius of a cylinder with the same height is called a hollow cylinder.
- Volume of a hollow cylinder \(=\) \(π(R^2−r^2)h\) cu. units.
Where \(r\) is the inner radius, \(R\) is the outer radius and \(h\) is the height.
Right circular cone:
A right circular cone is a cone whose apex (top vertex of the cone) is perpendicular to the centre of the base of the circle.
- Volume of a cone \(=\) \(\frac{1}{3}πr^2h\) cu. units.
Where \(r\) is the radius and \(h\) is the height.
Sphere:
A sphere is a three-dimensional figure obtained by the revolution of a semicircle about its diameter as an axis.
- Volume of a sphere \(=\) \(\frac{4}{3}πr^3\) cu. units.
Where \(r\) is the radius.
- Volume of a hollow sphere \(=\) \(\frac{4}{3}π(R^3−r^3)\) cu. units.
Where \(r\) is the inner radius and \(R\) is the outer radius.
Hemisphere:
A section of the sphere cut by a plane through any of its great circles is a hemisphere. In another way, we can say, one half of a sphere is called a hemisphere.
- Volume of a hemisphere \(=\) \(\frac{2}{3}πr^3\) cu. units.
Where \(r\) is the radius.
Hollow hemisphere:
A hemisphere emptied from the inner side and has a difference in the outer and inner radius of a hemisphere is called a hollow hemisphere.
- Volume of a hollow hemisphere \(=\) \(\frac{2}{3}π(R^3 − r^3)\) cu. units.
Where \(r\) is the inner radius and \(R\) is the outer radius.
Frustum of a cone:
If a smaller end of the cone is sliced by a plane parallel to its base, the portion of a solid between this plane and the base is known as the frustum of a cone.
- Volume of the frustum of a cone \(=\) \(\frac{1}{3}πh[R^2 + Rr + r^2]\) cu. units.
Where \(r\) is the inner radius, \(R\) is the outer radius and \(h\) is the height.