Cross-sectional area – Solid shapes – Formula

Cross-sectional area

The projected area after passing a plane through the 3-D object is said to be its cross-sectional area. Let us now find out the area of the cross-section for different solid shapes.

Sphere

When we cut a sphere along any plane passing through its centre, we get a circle having the same radius (r) as that of the sphere. The area of this circle is πr².

Therefore, the cross-sectional area of a circle having radius ‘r’ along any plane passing through the centre of the sphere is equal to πr².

Cube

When we cut a cube along a vertical plane or a horizontal plane, we get a square having the same sides ‘s’ as that of the cube. The area of this cube is equal to s^{2}.

Therefore, the Cross-sectional area of a cube having side ‘s’ along any vertical or horizontal plane is equal to s^{2}.

Cuboid

In the case of a cuboid having sides ( l × b × h ) when it is cut along the horizontal plane, we get a rectangle having sides (l × b). Here the area of this cross-section is (l × b).

Similarly, when we cut it through a vertical plane parallel to the front and back face, we get a rectangle having sides (l × h). Here the area of this cross-section is (l × h).

Again, when we cut it through a vertical plane parallel to the left and right sides, we get a rectangle having sides (b × h). Here the area of this cross-section is (b × h).

Cylinder

In the case of a cylinder having radius ‘ r ’ and height ‘ h ’, the area of the cross-section along the horizontal plane is equal to πr². Whereas the vertical cross-section passing through the centre of the cylinder is a rectangle having sides ( 2r (breadth), h (length) ) and its area is ( 2r × h ).

Right circular cone

The cross-section along a vertical plane passing through the tip of the cone is a triangle having base ‘2r’, height ‘ h ’. Here ‘ r’ and ‘ h ’ are the radius and height of the cone respectively. The area of this cross-section is equal to ‘ rh ’. 

The horizontal cross-section of a cone is a circle and its radius depends on the height of the cross-section from the base of the cone.

Solved Examples

1. Find the area of the cross-section for a cube having a side 4 cm along a vertical plane passed through it.

Solution:

s = Length of each side  = 4 cm 

Area of cross section =\mathrm{s}^{2}=4^{2}=16 \mathrm{~cm}^{2}

Hence, a vertical plane cuts the cube across a cross-section of 16 \mathrm{~cm}^{2}.

2. Find the area of cross-section of a right circular cone along a vertical plane passed through its tip. The height of this cone is 8 cm and the radius of its base is 2 cm.

Solution:

r = Radius of the base = 2 cm

h = Height = 8 cm

Area of cross-section =\mathrm{r h}=16 \mathrm{~cm}^{2}