What is the Perimeter of a closed shape?
Perimeter is the sum of the lengths of a shape that has edges and in case of shapes with curved lines, it’s the length of the boundary, which is considered as the perimeter. In simple words, we can say if we cut the boundary of a shape and make a straight line from that shape then the length of that straight line is the perimeter of that shape. The unit perimeter is the unit of length such as millimeter(mm), centimeter(cm), meter(m), kilometer(km), inch(in), yard(yd), feet(ft), and mile(mi).
What is the Area of a closed shape?
Area is the region covered by the boundaries of a closed geometric shape. In two dimensions we have shapes like triangle, quadrilateral, pentagon, circles and many more geometric shapes for which the area is calculated with specific formulas. The unit of area is the unit of (length)\(^2\) such as mm\(^2\), cm\(^2\), m\(^2\), km\(^2\), in\(^2\), yd\(^2\), ft\(^2\) and mi\(^2\).
Perimeter and Areas of Different Geometric Shapes
- Perimeter and Area of a Triangle
The perimeter of a triangle is the sum of all the three sides. The area of the triangle is half of the product of the base and height.
Perimeter of Triangle: P = a + b + c
Area of Triangle: A = \(\frac{1}{2}\)bh
- Perimeter and Area of a Rectangle
The perimeter of a rectangle is twice the sum of its length and width. The area of a rectangle is the product of its length times width.
Perimeter of Rectangle: P = 2(l + w),
Area of Rectangle: A = lw
- Perimeter and Area of a Parallelogram
The perimeter of a parallelogram is twice the sum of any pair of adjacent sides. The area of a parallelogram is the product of its base and height.
Perimeter of parallelogram: P = 2(a + b)
Area of Parallelogram: A = bh
- Perimeter and Area of a Square
The perimeter of a square is 4 times the length of one of the sides of the square. The area of a square is the product of the length of two of its sides.
Perimeter of Square: P = 4s
Area of Square: A = s\(^2\)
- Perimeter and Area of a Rhombus
The perimeter of a rhombus is 4 times the length of one of its sides. The area of a rhombus is half of the product of its diagonals.
Perimeter of Rhombus: P = 4s
Area of Rhombus: A = \(\frac{1}{2}d_1d_2\)
- Perimeter and Area of a Kite
The perimeter of the a is double the sum of the lengths of the pair of unequal sides.
Perimeter of Kite: P = 2(a + b)
The area is given by the relationship between the lengths of the parts of the diagonals.
Area of kite: A = \(\frac{1}{2}\)h(c + d) + \(\frac{1}{2}\)H(c + d), which also leads to the formula, A = \(\frac{1}{2}{\ d}_1d_2\)
- Perimeter and Area of a trapezoid
The perimeter of a trapezoid is the sum of all its sides.
Perimeter of Trapezoid: P = a + b + c + c or a + b + 2c
The area of a trapezoid is calculated by multiplying the average of the length of the parallel sides by its height.
Area of Trapezoid: A = \(\frac{1}{2}h(a+b)\)
- Perimeter (Circumference) and Area of a Circle
The perimeter or circumference of a circle is twice of pi times the radius. The area of a circle is pi times the square of radius.
Perimeter (circumference) of Circle: C = 2\(\pi\)r
Area of Circle: A = \(\pi\)r\(^2\)
Solved Examples
Example 1: Find the radius of a circle with a circumference of 176 cm.(Use \(\pi=\frac{22}{7}\))
Solution:
As stated, the circumference of a circle is 176 cm.
Radius of the circle can be calculated by using the circumference formula of the circle.
C = 2\(\pi\)r [Formula of circumference of circle]
176 = \(2\times\frac{22}{7}\times\)r [Substitute the value]
176 \(\times\)7 = 2 \(\times22\times\)r [Multiply both sides by 7]
r = \(\frac{176\times 7}{2\times 22}\) [Simplify]
r = 28 cm
So, the radius of the circle is 28 cm.
Example 2: Find the area of the rhombus whose diagonals are 14 cm and 8 cm.
Solution:
The diagonals of a rhombus are 14 cm and 8 cm. The area of a rhombus can be calculated by using the area formula of rhombus.
A = \(\frac{1}{2} d_1d_2\) [Area formula of Rhombus]
= \(\frac{1}{2}\times14\times8\) [Substitute the value]
= 56 cm\(^2\) [Simplify]
Hence, the area of the rhombus is 56 square centimeters.
Example 3: Sam was eating a sandwich that had a triangular shape, and an area of 44 cm\(^2\). The base of the triangle is 8 cm. Find the height of the sandwich.
Solution:
The area of triangular sandwich is 44 cm\(^2\) and its base is 8 cm.
The height can be calculated by using the area formula of a triangle.
A = \(\frac{1}{2}\)bh [Area formula of triangle]
44 = \(\frac{1}{2}\times8\times\)h [Substitute the value]
88 = \(8\times\)h [Multiply both sides by 2]
11 = h [Divide both sides by 2]
Or, h = 11
The height of the sandwich is 11 cm.
The perimeter is the total length that has been covered by the boundary of a closed shape.
The area of a closed shape is the surface of a closed geometric shape.
We know that in a regular polygon all the sides are equal, so its perimeter is n times the length of the length of a side of the regular polygon. In an irregular polygon all the sides are not equal, so to find the perimeter we have to add all the side lengths.
The radius of a circle is half the diameter of a circle.