Table Of Contents
- Cross Sections of Three Dimensional Figures
- Two Dimensional Shapes
- Three Dimensional Shapes
- Cross Sections of Three Dimensional Shapes
- Steps to Identify the Cross Sections
- Surface Area of Three Dimensional Objects
- Cross Section of a Cuboid
- Cross Section of a Prism
- Cross Section of a Cube
- Cross Section of a Cone
- Cross Section of a Cylinder
- Cross Section of a Pyramid
- Cross Section of a Sphere
- Solved Examples
- Frequently Asked Questions
Cross Sections of Three Dimensional Figures
Let us learn about the cross-sections of 3d figures by taking the example of a circular cake. When we cut the cake along the horizontal plane, the cake looks like a circle and when we cut the same cake towards the vertical axis, the cake looks like a rectangular box. The shapes that we see on cutting the cake in different ways are called cross-sections. It is very important to know about cross-sections to visualize the shapes of three-dimensional objects and convert them to two-dimensional objects.
Two-Dimensional Shapes
A two-dimensional shape is a flat plane figure or a shape with two dimensions in geometry, namely length and width. Two-dimensional (or 2-D) shapes have no thickness and can only be measured in two directions. We can categorize figures according to their dimensions. Two-dimensional shapes include the circle, triangle, square, rectangle and other polygons.
Three-Dimensional Shapes
A solid figure, object, or shape with length, width and height are referred to as three-dimensional shapes in geometry. Three-dimensional shapes have a thickness or depth, whereas two-dimensional shapes do not. Faces, edges and vertices are the characteristics of a three-dimensional figure. The 3D geometric shape’s edges are made up of three dimensions. The basic three-dimensional shapes that we see are a cube, a rectangular prism, a sphere, a cone, and a cylinder.
For example,
Cube: Rubik’s cubes or dice
Sphere: A baseball or a volleyball
Cone: Ice cream cones or carrots
Cylinder: Batteries or soda cans
Rectangular prism: Cinder blocks or books
Cross Sections of Three-Dimensional Shapes
- A two-dimensional slice formed by the intersection of a three-dimensional object with a plane is called a plane section.
- A cross-section is a plane section of a three-dimensional object that is perpendicular to one of the object’s lines of symmetry or parallel to one of its planes of symmetry.
- A copy of the base is any cross-section of such a figure cut with a plane parallel to the original plane.
The following are some examples of cross-sections for various shapes:
- A circle is the cross-section of a sphere.
- A triangle is the vertical cross-section of a cone, and a circle is the horizontal cross-section.
- A cylinder’s vertical cross-section is a rectangle, while its horizontal cross-section is a circle.
There are two types of cross sections. Those are:
a. Horizontal cross section
b. Vertical cross section
a. Horizontal cross-section: The horizontal cross-section of a solid figure is obtained by intersecting planes that are parallel to the solid figure’s base.
b. Vertical cross-section: The vertical cross-section of a solid figure is obtained by intersecting planes that are perpendicular to the solid figure’s base.
Steps to Identify the Cross Sections
Step 1: Determine the shape of the solid figure given.
Step 2: To make a horizontal cross-section cut the solid figure with a plane parallel to its base to get the desired shape.
(or)
Step 2: To make a vertical cross-section, cut the solid figure with a plane perpendicular to its base to get the cross-section of the shape.
Surface Area of Three Dimensional Objects
A three-dimensional object’s surface area is the sum of its two-dimensional surfaces (faces and bases).
Cross section of a Cuboid
A cuboid is a rectangular solid with six rectangular plane faces on all sides. A cuboid can be anything from a matchbox to a brick or even a book. A cuboid is made up of six rectangular faces, twelve edges and eight vertices.
- A horizontal plane passing through the lengths of the cuboid will cut it into two parts, each with a rectangular cross section.
- A vertical plane passing through the heights of the cuboid will cut it into two parts, each with a square cross section.
- The total surface area of a cuboid is the sum of the areas of all six faces.
Cuboid total surface area = 2(lb + bh + lh) sq. units
- The lateral surface area of a cuboid is the sum of the areas of its four walls.
Cuboid lateral surface area = [2h(l + b)]] sq. units
Cross section of a Prism
A prism is a polyhedron with two parallel and congruent faces, as well as parallelograms on the other faces. The bases of a prism are the two parallel and congruent faces of the prism. The lateral faces are the remaining parallelogram-shaped faces.
- A horizontal plane passing through the lateral sides of the prism will cut it into two parts, each with a triangular cross section.
- A vertical plane passing through the base of the prism will cut it into two parts, each with a rectangular cross section.
- The surface area of the prism is obtained by taking the sum of twice the area of the base and the lateral surface area of the prism.
Cross section of a Cube
A cube is a cuboid with the same length, width and height on all sides. A cube can be anything from ice cubes to sugar cubes to dice. The sides of each face are known as edges.
- A horizontal plane passing through the sides of the cube will cut it into two parts, each with a square cross section.
- A vertical plane passing through the top and bottom of the cube will cut it into two parts, each with a square cross section.
- Cube total surface area \(=(6a^2) sq. units\)
- Cube lateral surface area \(=(4a^2) sq. units\)
Cross section of a Cone
A cone is a three-dimensional geometric shape that narrows smoothly from a flat base (usually circular) to a point called the apex or vertex. The cone can also be defined as a pyramid with a circular cross-section, as opposed to a pyramid with a triangular cross-section. A circular cone is another name for these cones.
- A horizontal plane passing through the vertex of the cone will cut it into two parts, each with a triangle cross section.
- A vertical plane passing through the lateral sides of the cone will cut it into two parts, each with a circular cross section.
- A cone’s total surface area is equal to the sum of its base and lateral (side) surfaces.
The cone’s total surface area \(=T.S.A=\pi rl+\pi r^2\)
- The area of a cone’s lateral surface is the area of the lateral or side surface alone.
The cone’s lateral surface area = L.S.A=πrl , where l is the slant height.
Cross section of a Cylinder
Solids with a cylindrical shape include circular pillars, circular pipes, circular pencils, gauging jars, road rollers, gas cylinders etc. A right circular cylinder is a solid formed by the revolution of a rectangle around its sides in mathematical terms. Allow the rectangle ABCD to revolve around its side AB to form a right circular cylinder, as shown in the illustration. You’ve probably noticed that a right circular cylinder’s cross-sections are circles that are congruent and parallel to one another.
- A horizontal plane passing through the lateral side of the cylinder will cut it into two parts, each with a circular cross section.
- A vertical plane passing through the top and bottom ends of the cylinder will cut it into two parts, each with a rectangle cross section.
- Lateral Surface Area of a cylinder = 2πrh sq. units
- Total Surface Area of a cylinder = Curved Surface Area of cylinder + Area of the two circular ends \(=2\pi rh + 2r^2 sq. units.\) \(=2\pi r(h + r) sq. units\)
Cross section of a Pyramid
A pyramid is a polyhedron with a single base and the same number of triangular faces as its sides. The faces that are not at the bottom are referred to as lateral faces. A pyramid’s lateral faces are triangles.
- A horizontal plane passing through the lateral side of the pyramid will cut it into two parts, each with a square cross section.
- A vertical plane passing through the base side of the pyramid will cut it into two parts, each with a triangular cross section.
- A pyramid’s surface area is calculated by adding the areas of the base and each lateral face.
Cross section of a Sphere
A sphere is a collection of points in a three-dimensional space and they are all equidistant from the center. Spheres are geometric objects that are perfectly round. Polyhedra are not spheres. A circle is formed when a plane meets a sphere . A sphere’s cross-sections are all circles. (Each circle resembles the others).
- Any plane that intersects with a sphere at multiple points produces a cross section that is a circle.
- A sphere’s surface area is four times that of the greatest cross-sectional circle, known as the great circle.
Surface Area\(=4 \pi r^2=\pi d^2\)
Where, r = sphere radius.
d = sphere diameter.
Solved Examples
Example 1: Identify the intersection of the plane and the solid.
a. 
b. 
c. 
d. 
e. 
Solution:
a. The intersection of a plane and a triangular pyramid is depicted in this diagram. The intersection is a trapezium.
b. The intersection of a plane and a triangular prism is depicted in this diagram. The intersection is a triangle.
c. The intersection of a plane and a cylinder is depicted in this diagram. The intersection is a rectangle.
d. The illustration depicts the meeting of a plane and a cone. The intersection is a circle.
e. The intersection of a plane and a cylindrical pipe is depicted in this diagram. The intersection is a circle.
Example 2: Determine whether the intersection in question is feasible. If that’s the case, make a solid and a cross section. A circle is formed when a plane and a sphere intersect.
Solution: Yes, the given intersection is possible to draw. The intersection of a plane and a sphere is a circle.
This illustration shows the intersection of a plane and a sphere. The intersection is a circle.
Example 3: The cylindrical water tanker has a height of 220 centimeters and a diameter of 40 centimeters. Calculate the cylindrical tank’s curved surface area and total surface area.
Solution: Given data, height of the cylinder, (h) = 220 cm
Diameter of the tank (d) = 40 cm
Therefore, the radius of the base (r) \(=\frac{40}{2}=20~cm\)
Curved surface area of the cylindrical water tank \(=2\pi rh\)
\(=2\times \frac{22}{7}\times 20\times 220\) (Multiplying)
\(\approx 27657~sq.cm\)
Therefore, the curved surface area of the cylindrical water tank is 27657 sq.cm.
Total surface area of the cylindrical water tank = 2πr(h + r)
\(=\left[2\times \frac{22}{7}\times 20(220 + 20)\right]\) (Multiplying)
\(\approx 30171~sq.cm\)
Therefore, the total surface area of the cylindrical tank is 30171 sq.cm.
When you cut straight through an object, the shape you get is called the cross-section. Depending on how it was cut, it could be a rectangle, a circle or even an oval.
If there are any axes in the solid, cross sections can be perpendicular to their orientation. Any splitting plane through a sphere, regardless of orientation, will produce a disc of some size. The area of cross-section is determined by the solid’s shape, which determines the boundaries, the cross-sections as well as the angle between the solid’s axis of symmetry and the plane that produces the cross-section.