Perimeter and Area
The word perimeter comes from two Greek words “Peri” and “metron”. The word “peri” means around and “metron” indicates the measurement. It was first used in the 15th century. The perimeter of any geometric figure refers to the total length of the sides or edges of the figure.
The term area indicates the quantity of space enclosed by a 2D shape. It is the measurement of the surface of any geometrical figure. The measurement of the area first started from the ancient city of Babylon. It was this measurement of area that was employed there to indicate the land owned by different people. As a consequence, this measurement of area helped officials to evaluate taxation of the public.
In 287 BC, the great mathematician Archimedes of Greece discovered and proved the formula for the perimeter and area of a circle.
What is Area?
Area is the quantity of space a flat shape takes up on a plane. If we take the example of a wall in a room then the area is the region of the wall that can be painted using a color of our choice. Generally, it is denoted by “A”.
In any closed figure, the area indicates the number of unit squares that covers the whole surface. It is measured in square units such as square inches, square feet, etc.
What is Perimeter?
The perimeter of any geometrical figure is the distance along its outer edges. The geometrical figures should be a closed one at that if we want to measure the perimeter. The total length of all the sides or edges of a polygon indicates the perimeter of a polygon.
The perimeter of any two-dimensional geometrical figure is the length of the boundary of the shape. It is denoted by “P”.
For example the perimeter of the irregular hexagon in the image is,
P = 5 + 1 + 3 + 3 + 4 + 2 = 18 inches
Perimeter and Area Formulas:
Formula for the perimeter of a triangle given the three sides:
Perimeter of triangle (P) = a + b + c unit.
Perimeter of special triangles-
Formula for the perimeter of an equilateral triangle:
Equilateral triangle perimeter (P) = 3 × side = 3a units.
Where “a” is the side of the equilateral triangle.
Formula for the perimeter of an isosceles triangle:
Isosceles triangle perimeter (P) = 2a + b units.
Where “a” is the length of equal sides and “b” is unequal.
Area of triangle formula (A) = \( \frac{1}{2}\) × base × height Sq. units
Area of Isosceles triangle formula A = \(\sqrt{s(s-a)(s-b)(s-c)}\) Sq. units.
This area of triangle formula is known as Heron’s formula.
Area of special triangles-
Formula for the area of equilateral triangles:
Equilateral triangle Area A = \(\frac{\sqrt{3}}{4}a^2\) Sq. units
Where “a” is the side of the equilateral triangle.
Formula for the perimeter of a quadrilateral:
Perimeter of any quadrilateral whose four sides are a,b,c and d is given by,
Quadrilateral perimeter (P) = a + b + c + d units.
Formula for the perimeter and area of a rectangle:
Perimeter of a rectangle
(P) = 2 l + 2 w units.
Where “l” is length and “w” is the width of a rectangle.
Area of Rectangle (A) = l × w Sq. units
Where “l” is length and “w” is the width of a rectangle.
Formula for the perimeter and area of a square:
Perimeter of a square (P) = 4 a units.
Where “a” is the side of the square.
Area of a square (A) = \(a^2\) Sq. units.
Formula for the perimeter of a circle ( also referred to as the circumference of a circle):
Perimeter of a circle C = 2\(\pi\)r units.
Area of a circle A = \(\pi r^2\) square units
Where “r” is the radius of the circle.
Formula for the perimeter of a semicircle:
Perimeter of a semi-circle = \(\pi\)r + 2r units.
Area of a semi-circle = \(\pi r^2\)/2 unit.
Where “r” is the radius of a circle and “d” is the diameter of a semicircle circle.
Figure | Perimeter | Area |
|---|---|---|
Triangle | a + b + c | \(\frac{1}{2}\times b\times h\) |
Rectangle | 2l + 2w | l × w |
Square | 4a | \(a^2\) |
Circle | 2πr | \(\pi r^2\) |
Solved Examples:
Example 1: Find the perimeter and area of a square with each side measuring 5 inches.
Solution:
The perimeter of a square (P) = 4 × a
= 4 × 5 [Replace a with 5]
= 20 inches.
The area of square (A) = \(a^2\) Sq. inches
= \(5^2\) [Replace a with 5]
= 25 Sq. inches [Simplify]
Perimeter of the square is 20 inches and the area is 25 square inches.
Example 2: Find the perimeter and area of a rectangle of length 12 inches and width 10 inches.
Solution:
The perimeter of a rectangle (P) = 2l + 2w
= 2 × 12 + 2 × 10 [Replace l with 12 and w with 10]
= 24 + 20 [Simplify]
= 44 inches [Simplify]
The area of a rectangle (P) = l × w square inches
= 12 × 10 [Replace l with 12 and w with 10]
= 120 square inches [Simplify]
Perimeter of the rectangle is 44 inches and the area is 120 square inches.
Example 3: Find the perimeter and area of an equilateral triangle with each side measuring 12 feet.
Solution:
The perimeter of a equilateral triangle P = 3a [Perimeter of equilateral triangle formula]
= 3(12) [Replace a with 12]
= 36 feet [Simplify]
Perimeter of the equilateral triangle is 36 feet.
The area of equilateral triangle (A) = \(\frac{\sqrt{3}}{4}a^2\) [Area of a equilateral triangle formula]
= \(\frac{\sqrt{3}}{4}(12^2)\) [Replace a with 12]
= 63.352 square feet [Simplify]
Perimeter of the triangle is 36 feet and the area is 63.352 square feet.
Example 4: Find the circumference and area of a circle of radius measuring 10 feet.
Solution:
Circumference of circle = \(2\pi r\) [Circumference of circle formula]
= 2\(\pi\)(10) [Replace r with 10]
= 2 x 3.14 x 10 [Replace \(\pi \) with 3.14]
= 62.8 feet [Simplify]
Area of circle (A) = \({\pi}r^2\) [Area of circle formula]
= \({\pi}\times10^2\) [Replace r with 10]
= 3.14 × 100 [Replace pi with 3.14]
= 314 square feet [Simplify]
The circumference of circle is 62.8 feet and area of the circle is 314 square feet.
Example 5: Larry has bought a plot in Chicago. The plot is in the shape of a rectangle. The length of the plot is 200 feet. He walked along its boundary and observed that he had walked a total distance of 600 feet. Can you calculate the width of the plot?
Solution:
Luci has walked through the boundary of the plot. This means he walked equal distance to the perimeter of the plot. So,
P = 2l + 2w [Perimeter of a rectangle formula]
600 = 2 x 200 + 2w [Replace P with 600 and l with 200]
600 = 400 + 2w [Simplify]
200 = 2w [Subtract 400 on each side]
100 = w [Divide each side by 100]
The width of the rectangular plot is 100 feet.
Example 6: If the radius of a circular swimming pool is 25 feet, then, find the circumference of the pool.
Solution:
Circumference of the swimming pool = 2\(\pi\)r [Circumference of circle formula]
= 2\(\pi\)(25) [Replace r with 25]
= 2 x 3.14 x 25 [Replace \(\pi\) with 3.14]
= 157 feet [Simplify]
The circumference of the swimming pool is 157 feet.
No, for any given figure there will be just one unique perimeter.
Perimeter of a triangle given two sides and an angle between them is:
\(P = \sqrt{a^2+b^2+2ab\cos\theta}\)
Where a, b are sides of the triangle and is the angle between them.
The perimeter of a regular hexagon is 6a units. Where “a” is the length of the side of the regular hexagon.
Divide the quadrilateral into two triangles using any diagonal. Then measure the lengths of the sides and use Heron’s formula to find the area of both triangles. At last add the areas of triangles to find the area of the quadrilateral.