Written By Priya_Singh
Last Modified 22-06-2023

Plane Figures and Solid Shapes: Properties, Formulas

Plane Figures and Solid Shapes: In geometry, shapes are the forms of objects which have boundary lines, angles and surfaces. These figures demonstrate the shape of objects we see in our everyday lives. There are different types of 2−D shapes and 3−D shapes. The plane shapes are two-dimensional closed shapes with no thickness and are known as 2−D shapes.

Solid shapes are nothing but solids with three dimensions: length, breadth, and height. Solid shapes are also known as 3−D shapes. This article discusses the concepts of Plane Figures and Solid Shapes in detail. Read on to find out.

What are Plane Figures and Solid Shapes?

Plane figure: A shape can be defined as the boundary or outline of an object. A plane shape is a two-dimensional closed figure that has no thickness. A plane in geometry is a flat surface that extends up to infinity in all directions. A plane figure is a geometric figure having no thickness. It may consist of line segments, curves or a combination of both line segments and curves.

Examples: Few of the plane shapes are square, circle, rectangle, triangle etc.

Solid figure: Solid shapes are three-dimensional figures that have length, breadth, and height as the three dimensions.

Examples: Few of the solid shapes are cube, cuboid, sphere etc.

Difference Between Plane Figures and Solid Figures

The difference between the plane shapes and the solid shapes is given below:

Plane Shapes	Solid Shapes
Plane shapes have only two dimensions: length and width.	Solid shapes have three dimensions: length, width and height.
Plane shapes are square, triangle, circle, rectangle, hexagon, etc.	Solid shapes are sphere, cube, cuboid, cylinder, etc
Plane shapes involve length and breadth.	Solid shapes involve length, breadth and height.
Plane shapes are straightforward to construct.	Solid shapes are quite complex to construct.
The edges of the plane shapes are evident when you draw them. Example: Edges are visible in a rectangle.	All the edges of the solid shapes are not visible due to overlapping. Example: All the edges of the cube are not displayed from one angle.

Types of Plane and Solid Figures

The different types of plane figures are given below:

Square
Rectangle
Triangle
Circle
Pentagon
Octagon

Square

The square is the two-dimensional plane shape that has four equal sides and four vertices. All the angles in the square are equal to \({\rm{9}}{{\rm{0}}^ \circ }\). The diagonals of the square also bisect each other at \({\rm{9}}{{\rm{0}}^ \circ }\). This shape is also considered a two-dimensional regular quadrilateral.

Example: The wall or the table that has all four equal sides are examples of a square.

Rectangle

The rectangle shape is the two-dimensional plane shape whose opposite sides are equal and parallel to each other. All the angles in the rectangle shape are equal to \({\rm{9}}{{\rm{0}}^ \circ }\).

Example: Blackboards, dollars note, currency notes etc., are examples of rectangular shapes.

Triangle

The triangle shape is a three-sided polygon that has three sides and three vertices. We have different types of triangles that are shown below in the diagram. The sum of all the three angles in the triangle is equal to \({\rm{18}}{{\rm{0}}^ \circ }\).

Example: Pyramids are one of the best examples of triangle shapes.

Circle

The circle is a two-dimensional plane shape that has only a curved boundary. The distance from the centre point to the outer line of the circle is known as the radius.

Examples: Bangles, pizza, wheels of the vehicles, and orbit are a few examples of a circle shape.

Pentagon

The pentagon is a \(2D\) plane shape. It is a five-sided polygon, and the pentagon can be regular or irregular. In the regular pentagon, each interior angle is equal to \({\rm{10}}{{\rm{8}}^ \circ }\), and each exterior angle measures \({72^ \circ }\). This shape has five diagonals.

Example: The US Department of Defence headquarters has the shape of a Pentagon. It is a perfect example of the pentagon shape.

Octagon

An octagon is a polygon with eight sides and can be either regular or irregular. It is a \(2D\) shape with eight angles, and the sum of all interior angles of an octagon is \({\rm{108}}{{\rm{0}}^ \circ }\).

Example: The stop signboard that you can see on the roadside has an octagon shape.

Solid Shapes

The different types of solid shapes are given below:

Cube
Cuboid
Sphere
Cylinder
Cone

Cube

The cube is a three-dimensional solid shape with \(6\) faces of square shape, \(8\) vertices, and \(12\) edges.

Examples: Rubric cube, dice, sugar cubes, gift box etc., are a few of the examples of the cube.

Cuboid

The cuboid is a three-dimensional solid shape with \(6\) faces, \(8\) vertices and \(12\) edges. All the angles measure \({90^ \circ }\)

Examples: Lunchboxes, shoebox, bricks, mattresses, books etc., are a few examples of the cuboid.

Sphere

The sphere is a three-dimensional solid shape that is perfectly round, and all the points on its surface are equidistant from the centre point. The distance from the centre point to the outer line of the sphere is known as the radius. The sphere has \(1\) curved face, \(0\) edges and \(0\) vertices.

Examples: Ball, planets, orange, marbles, eyeballs etc., are a few examples of the sphere.

Cylinder

A cylinder is a three-dimensional solid shape that has \(2\) circular bases connected to the curved surface. It has \(0\) vertices, \(2\) flat circular faces, \(1\) curved face and \(2\) edges.

Examples: Water cans, batteries, candles, beakers, test tubes, etc., are a few examples of the cylinder.

Cone

A cone is a three-dimensional object or solid shape with a circular base and \(1\) vertex. The cone is the figure that decreases smoothly from the circular base to the point known as the apex. The cone has \(1\) vertex, \(1\) edge, \(1\) flat circular face and \(1\) curved face.

Examples: Funnels, party hats, traffic cones, Christmas trees, ice cream cones etc., are a few of the examples of the cone.

Area and Perimeter of 2D Shapes

\(2D\) Shapes	Area	Perimeter
Circle	\(\pi {r^2}\) is the radius of the circle)	\(2\pi r\)
Triangle	\(\frac{1}{2}\left( {{\rm{Base}} \times {\rm{height}}} \right)\)	Sum of the sides
Square	\({\rm{Sid}}{{\rm{e}}^{\rm{2}}}\)	\(4\left( {{\rm{Side}}} \right)\)
Rectangle	\({\rm{Length}} \times {\rm{Breadth}}\)	\(2\left( {{\rm{Length}} + {\rm{Breadth}}} \right)\)
Rhombus	\(\frac{1}{2}\) (Product of diagonals)	\(4\left( {{\rm{Side}}} \right)\)
Parallelogram	\({\rm{Base}} \times {\rm{Height}}\)	\(2\left( {{\rm{Base}} + {\rm{Height}}} \right)\)

Surface Area and Volume of 3D Shapes

Shapes	Surface Area	Volume
Cube	The surface area of cube \( = 6{a^2}\) square units	The volume of cube \( = {a^3}\) cubic untis
Cuboid	The surface area of cuboid \( = 2\left( {lb + bh + lh} \right)\) square units	The volume of a cuboid \( = lbh\) cubic units
Cylinder	The surface area of the cylinder \( = 2\pi r\left( {h + r} \right)\) square units The curved surface area of a cylinder \(= 2\pi rh\)	The volume of cylinder \(= \pi {r^2}h\) cubic units
Cone	The surface area of a cone \( = \pi r\left( {r + \sqrt {\left( {{r^2} + {h^2}} \right)} } \right)\) square units THe curved surface area of a cone \( = \pi rl\) square units The slant height of a cone \( = l\sqrt {\left( {{r^2} + {h^2}} \right)} \) units	The volume of cone \( = \frac{1}{3}\pi {r^2}h\) cubic units
Sphere	The total surface of a sphere \(= 4\pi {r^2}\) square untis	The volume of sphere \( = \frac{4}{3}\left( {\pi {r^3}} \right)\)
Pyramid	The surface area of a pyramid \( = \left( {{\rm{base}}\,{\rm{area}}} \right) + \left( {\frac{1}{2}} \right) \times \left( {{\rm{perimeter}}} \right) \times \left( {{\rm{slant}}\,{\rm{height}}} \right)\) square units	The volume of a pyramid \( = \frac{1}{3} \times \left( {{\rm{base}}\,{\rm{area}}} \right) \times {\rm{height}}\) cubic units

Solved Examples – Plane Figures and Solid Figures

Q.1: Find the volume and surface area of a cuboid of l=10 cm, b=8 cm and h=6 cm
Ans: We have the volume of cuboid \(=376 \mathrm{~cm}^{2}\)
\( = 10 \times 8 \times 6\)
\( = 480\;{\rm{c}}{{\rm{m}}^2}\)
Surface area \( = 2(lb + lh + bh)\)
\( = 2(10 \times 8 + 10 \times 6 + 8 \times 6)\)
\( = 2(80 + 60 + 48)\)
\( = 376\;{\rm{c}}{{\rm{m}}^2}\)

Q.2: The length of the rectangular field is 15 m , and the width is 7 m. Find the area and perimeter of the field.
Ans: Given,
Length \(=44 \mathrm{~m}\)
Width \({\rm{ = 7m}}\)
We have, area formula \({\rm{A = length \times width}}\)
\(=15×7\)
\( = 105\;{{\rm{m}}^2}\)
And perimeter formula \(P = 2({\rm{length + width}})\)
\(=2(15+7)\)
\(=2×22\)
\( = 44\;{\rm{m}}\)

Q.3: Find the perimeter of the rectangle whose length and breadth are 10 cm and 6 cm, respectively. Also, find its area.
Ans: Given,
Length of the rectangle \({\rm{ = 10cm}}\)
The breadth of the rectangle \({\rm{ = 6cm}}\)
Area of rectangle \({\rm{ = length \times breadth}} = 10 \times 6 = 60\;{\rm{c}}{{\rm{m}}^2}\)
Perimeter of rectangle \( = 2({\rm{length + breadth)}}\)
\( = 2(10 + 6) = 2 \times 16 = 32\;{\rm{cm}}\)

Q.4: What is the area of a square that has a side length equal to 4 inches?
Ans: Given that the length of the side of square \(=4\) inches.
Area of square \( = {{\mathop{\rm side}\nolimits} ^2} = {(4)^2} = 16{\rm{inc}}{{\rm{h}}^2}\)
Hence, the required answer is \(16{\rm{inc}}{{\rm{h}}^2}\)

Q.5: What is the area of the circle whose radius is 7 cm? \(\left( {\pi = \frac{{22}}{7}} \right)\)
Ans: Given,
The radius of circle \(=7 \mathrm{~cm}\)
Area of circle \(=\pi r^{2}=\left(\frac{22}{7}\right) \times 7^{2}\)
\(=22 \times 7=154 \mathrm{sq.cm}\)

Summary

In this article, we have discussed plane figures and solid shapes with examples. Then we discussed the difference between plane figures and solid figures. We glanced at the similarities between plane figures and solid figures followed by plane figures like square, triangle, rectangle, & circle and types of solid figures like the cube, cuboid, sphere, etc. Finally, we have listed some solved examples along with a few FAQs.

Frequently Asked Questions (FAQs) – Plane Figures and Solid Shapes

Q.1: What is a flat surface of a solid figure called?
Ans: The flat surface is known as the face of the solid. The edge is formed when two solid faces meet. So, you can say that the flat surface present on any solid is considered to be a face.

Q.2: What are the plane figures and solid figures?
Ans: Plane figure: A plane shape is a two-dimensional closed figure that has no thickness. A plane in geometry is a flat surface that extends up to infinity in all directions. It has infinite width and length and zero thickness.
Solid figure: Solid shapes are three-dimensional figures that have length, breadth, and height as the three dimensions.

Q.3: What are the differences between plane shapes and solid figures?
Ans: The difference between the plane shapes and the solid shapes is given below:

Plane Shapes	Solid Shapes
Plane shapes have only two dimensions: length and width.	Solid shapes have three dimensions: length, width and height.
Plane shapes are square, triangle, circle, rectangle, hexagon, etc.	Solid shapes are sphere, cube, cuboid, cylinder, etc
Plane shapes involve length and breadth.	Solid shapes involve length, breadth and height.
Plane shapes are straightforward to construct.	Solid shapes are quite complex to construct.
The edges of the plane shapes are evident when you draw them. Example: Edges are visible in a rectangle.	All the edges of the solid shapes are not visible due to overlapping. Example: All the edges of the cube are not displayed from one angle.

Q.4: What are examples of plane figures and solid figures?
Ans: The examples of the plane figures and the solid figures are given below:
Plane figures: square, rectangle, triangle, circle, pentagon, octogen, hexagon etc., are examples of plane figures.
Solid figures: cube, cuboid, cylinder, sphere, cone, pyramid, prism etc., are examples of solid figures.

Q.5: How do you teach solid shapes?
Ans: Solid shapes have three dimensions: length, width and height. They occupy volume in space. Their surfaces are known as faces. You can say that the faces meet at the edges. A few examples of types of solid shapes are cone, cuboid, sphere, cube and cylinder.