# Surface Areas and Volumes

### Cube and Cuboid

A cuboid is a solid bounded by six faces that are rectangular in shape. A cuboid whose length, breadth and height are equal is called a cube.

Any two faces other than the opposite faces are called adjacent faces. Any two adjacent faces meet in a line segment, which is called an edge of the cuboid. The point of concurrency of any three edges of a cuboid is called a vertex of the cuboid.

A cube and a cuboid have 8 vertices, 6 faces and 12 edges. Any face of a cuboid may be called the base of the cuboid. In that case, the four adjacent faces of the base are called the lateral faces of the cuboid. Total surface area of a solid is equal to the sum of the areas of all its faces. Area is measured in square units.

Volume of a solid is the measure of the space occupied by it. The volume of substance that can be stored by the object is called its capacity. Volume is measured in cubic units.

If l, b and h are the length, breadth and height of a cuboid respectively, then
•  lateral surface area = 2h (l + b)
•  total surface area = 2(lb + bh + lh)
•  volume = lbh

If 'a' is the length of the edge of a cube, then
•  lateral surface area = 4a2
•  total surface area = 6a2
•  volume = a3

### Cylinder

A cylinder can be defined as a solid figure that is bound by a curved surface and two flat circular surfaces. The flat surfaces are made up of two congruent circles that are parallel to each other. These flat surfaces are called the bases of the cylinder. The radius of the circular bases is the radius of the cylinder. The perpendicular line that passes through the centres of the two circular bases is the height of the cylinder or axis of the cylinder. A cylinder is said to be right circular cylinder if axis of the cylinder is perpendicular to the radius of the cylinder.

Curved surface area of a cylinder (CSA)

The curved surface joining the two bases of a right circular cylinder is called its lateral surface. If the curved surface of a cylinder is unfold, it opens up to become a rectangular sheet. The curved surface area of a cylinder is equal to the area of this sheet.
Area of a rectangle = length x breadth.
The length of the rectangular sheet is equal to the circumference of the base of the cylinder, which is equal to 2πr and the breadth of the rectangular sheet is equal to the height of the cylinder.
Substituting length = 2πr and breadth = h, we get
Area of the rectangular sheet = 2πr × h = 2πrh
∴ Curved surface area of a cylinder (CSA) = 2πrh.

Total surface area of a cylinder

Total surface area of a cylinder (TSA) = Curved surface area (CSA) + Total area of the flat surfaces
The flat surfaces of a cylinder are made up of two congruent circles.
∴ Area of the flat surface of a cylinder = Area of the base = πr2
Total area of both the flat surfaces = 2πr2
Total surface area of a cylinder (TSA) = 2πrh + 2πr2 = 2πr(h + r).

Volume of a cylinder

Volume of a cylinder = Area of the base × height
= πr2 × h (∵ Area of the base = πr2)
∴ Volume of a cylinder (V) = πr2h.

### Cone

A cone is a solid figure with a circular base that tapers to a point called the vertex. It is said to be a right circular cone if the line joining the vertex of a cone to the centre of its base is at right angle to the radius of the base.

Let 'r' be the base radius, 'h' be the height, 'l' be the slant height of a right circular cone. Slant height, radius and the height of a cone forms a right angled triangle. Then by applying Pythagoras theorem l = .

Base area of the cone = πr2

Curved surface area of the cone = πrl

Total surface area of the cone = πrl + πr2 = πr(l + r)

Volume of the cone = πr2h

### Sphere and Hemisphere

A sphere is a three dimensional figure, made up of points that are equidistant from a given point. It does not have an edge or a vertex. The surface of a sphere is uniform and smooth. The centre of a sphere is a point, which is equidistant from all the points on the sphere.
The distance between the centre and any point on the surface of the sphere is called the radius of the sphere. Generally the radius is denoted by the letter r.

A line segment through the centre of the sphere, and with the end points on the sphere is called a diameter of the sphere.

Surface area of a sphere (A) = 4πr2

Volume of a sphere (V) =  πr3

A plane through the centre of a sphere divides the sphere into two equal parts, each of which is called a hemisphere.

A hemisphere has two faces. A flat surface which is called as the base and a curved surface.

Surface area of a hemisphere = 2πr2

Total surface area of a hemisphere = 3πr2

Volume of a hemisphere = πr3

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