What are Shapes?
Shapes, or geometric shapes are figures with outer boundaries that are created by combining curves, points, and lines. A shape can be an open shape or a closed shape.
As you can see, in the first image, the line segment of the shape does not meet, whereas in the second image, the shape is a closed figure with connected curves.
Closed shapes can be further classified into two types:
- Two-dimensional shapes
- Three-dimensional shapes
What are Two-Dimensional Shapes?
Two-dimensional or 2D shapes are flat plane figures with two dimensions — length and width. There is no thickness in 2D shapes. Some examples of two-dimensional shapes are circles, triangles, squares, rectangles, and pentagons.
What are Three-Dimensional Shapes?
Three-dimensional or 3D shapes are solid figures or objects that have three dimensions — length, width, and height. Some examples of three-dimensional shapes are cubes, rectangular prisms, spheres, cones, and cylinders.
What is the definition of Area?
Area is defined as the measure of space within the boundary of a shape. Area is usually represented by the number of square units, such as square centimeters, square feet, square inches, and so on.
What is the Area of 2D Shapes?
The area of any 2D shape is given by the measure of space enclosed within its boundary.
The formula used to calculate the area of a 2D shape depends upon the type of 2D shape.
Area of square
Area = Side x Side
= s x s, where s is the length of the side of the square.
Area of a rectangle
Area = Length x Width
= l x w, where l is the length and w is the width of the rectangle.
Area of a triangle
Area = \( \frac{1}{2}\times b \times h \), where b is the base and h is the height of the triangle.
Area of a circle
Area = \( \pi \times r^2 \), where r is the radius of the circle.
Area of a parallelogram
Area = b x h, where b is the base and h is the vertical height.
Area of a trapezoid
Area = \( \frac{1}{2}(a+b)\times h \), where a and b are the lengths of the parallel sides and h is the height or the perpendicular distance between parallel sides.
What is the Area of 3D Shapes?
A 3D shape consists of multiple surfaces, so the area of a 3D shape is the sum of the areas of these surfaces. Hence, the area in 3D shapes is known as the surface area.
The formula used to determine the area of a 3D shape depends upon the type of 3D shape.
Surface area of cube
Surface area = 6a\( ^2 \), where a is the length of the edge of the cube.
Surface area of a cuboid or rectangular prism
Surface area = 2(lh + wh + lw), where l is the length, h is the height, and w is the width of the cuboid.
Surface area of a cylinder
Surface area = 2πr(r + h), where r is the radius of the circular base and h is the height of the cylinder.
Surface area of a sphere
Surface area = 4πr\( ^2 \), where r is the radius of the sphere.
Surface area of a cone
Surface area = πr(r + l), where r is the radius of the circular base and l is the slant height of the cone.
Solved Examples on Area of Shapes
Example 1: Find the area of a circular garden whose radius is 14 m.
Solution: Here, the radius of the circular garden is 14 m.
The formula for the area of a circle is given by \( \pi r^2 \) and \( \pi \) is \( \frac{22}{7} \).
So, Area \( =\pi r^2 \)
\( ~~~~~~~~~~~~~~=\frac{22}{7}\times 14\times 14 \) [Substitute values]
\( ~~~~~~~~~~~~~~=616 \) sq.m. [Simplify]
Hence, the area of the circular garden is 616 sq.m.
Example 2: Find the surface area of a cone if the radius is 6 cm and the slant height is 4 cm.
Solution: Here, the radius is 6 cm and the slant height is 4 cm.
As per the formula, the surface area of a cone \( =\pi r(r+l) \)
Surface area \( =\pi \times 6(6+4) \)
\( ~~~~~~~~~~~~~~~~~~~~~=\frac{22}{7}\times 6\times 10 \) [Substitute values]
\( ~~~~~~~~~~~~~~~~~~~~~=188.57 \) cm\( ^2 \) [Simplify]
Hence, the surface area of the cone is 188.57 cm\( ^2 \).
Example 3: Alice bought a cuboid-shaped aquarium. Its dimensions are 6 cm, 4 cm, and 3 cm. What is the surface area of the aquarium?
Solution: Here, the length of the aquarium is 6 cm, the height is 4 cm, and the width is 3 cm. So, the surface area of the aquarium is given by \( 2(lh + wh + lw) \).
Surface area \( =2(lh + wh + lw) \)
\( ~~~~~~~~~~~~~~~~~~~~~= 2 (6 \times 4 + 3 \times 4 + 6 \times 3) \) [Substitute values]
\( ~~~~~~~~~~~~~~~~~~~~~= 2 (24 + 12 + 18) \) [Simplify]
\( ~~~~~~~~~~~~~~~~~~~~~= 2 \times 54 \) [Simplify further]
\( ~~~~~~~~~~~~~~~~~~~~~= 108 \) cm \( ^2 \) [Multiply]
The surface area of the aquarium is 108 cm\( ^2 \).
A 2D shape is drawn in two dimensions — length and width. In comparison a 3D shape is drawn using three dimensions — length, width, and height or thickness.
A solid shape is a three-dimensional object that has three dimensions.
A polygon is a closed two-dimensional shape bounded by straight lines. Examples: triangle, square, rhombus, trapezium, pentagon, and so on.
A quadrilateral is a type of polygon with exactly four sides. Examples: square, rectangle, parallelogram, and so on.
Area is the measure of space occupied by a flat surface in a 2D plane, whereas surface area is the sum of the measures of the area of the surfaces exposed in a 3D shape.
If you see that your child is struggling with a particular concept or operation that you thought they had mastered, a math worksheet is a great way to review previously practiced skills.