In geometry, we study mainly two dimensions: two-dimensional objects and three-dimensional objects. The two-dimensional objects are plane figures that can be represented only in two directions. In contrast, three-dimensional objects can be expressed in three directions.
Strictly speaking, if we add a third side on two-dimensional objects, then they become three-dimensional. For example, a square is a two-dimensional figure with sides, length and breadth. If we add a third side, height, then it becomes a cube. However, the third side must be of equal length as that of the other two sides to form a cube.
The area is defined as the region bounded within the sides of a polygon. It is a two-dimensional quantity. It expresses the extent in a planar lamina. Lamina includes all 2D shapes, i.e. square, rectangle, triangle, circle, trapezium, etc. For example, if a body has four sides, then the inside region will denote its area.
Shape | Area | Terms |
Circle | π × r² | r = radius of the circle |
Triangle | ½ × b × h | b = base h = height |
Square | a² | a = length of side |
Rectangle | l × w | l = length w = width |
Parallelogram | b × h | b = base h = vertical height |
Trapezium | ½ (a + b) × h | a and b are the length of parallel sides h = height |
Ellipse | π a b | a = ½ minor axis b = ½ major axis |
The surface area is the outside area of a 3D shape. The space inside a 3D object will be its capacity, referred to as volume. Therefore, only the outside area of the surface will be considered in calculating the area.
For example, take a hollow ball. The outer shell will represent its surface area, whereas the inside hollowness will represent its volume. Thus, the surface area is dominant in 3D shapes only. Some 3D shapes include cubes, cuboids, spheres, prisms, cones, hemispheres, etc.
Shape | Surface Area | Terms |
Cube | 6a² | a = length of the edge |
Rectangular prism | 2 (wl + hl + hw) | l = length w = width h = height |
Cylinder | 2πr (r + h) | r = radius of circular base h = height of the cylinder |
Cone | π r (r + l) | r = radius of circular base l = slant height |
Sphere | 4πr² | r = radius of sphere |
Hemisphere | 3πr² | r = radius of hemisphere |
Area | Surface Area |
It is a two-dimensional property. | It is a three-dimensional property. |
It is the space occupied by lines of a shape. | It is the outer area of any body, not the inside space. |
Examples of shapes having area are - square, rectangle, triangle, circle, etc. | Examples of shapes having surface area are cubes, cuboids, prisms, cones, etc. |
We can calculate area by knowing the sides of any 2D shape. | We can calculate the surface area if we know all the sides of any shape. |
Real life applications of area include finding the quantity of paint required to paint a wall, sheet of paper required to draw a book, etc. | Real life applications include finding the quantity to paint a matchbox, gift wrap required to wrap a book, etc. |
Example 1: Find the area and surface area of a circular park whose radius is 9 m.
Solution:
From the formula, we know,
Area of circle = π r²
A = 22/7 x 9 x 9
A = 254.57 sq.m.
Also, we know the surface area of a sphere = 4πr²
A = 4 x 22/7 x 92 = 1018.28 sq.m.
Example 2: Find the area of the cone, whose radius is 7 cm and height is 5 cm.
Solution:
We know, the area of cone = π r (r + l)
Where, l = slant height of the cone, and is given by:
l = √(7² + 5²) = √49 + 25 = 8.6 cm
Therefore, Area = (22/7) × 7 (7 + 8.6)
= 22/7 x 7 x 15.6 = 343.2 cm²