When we study geometry, two important measures come again and again: Perimeter and Area.
Perimeter is the distance around a closed figure. Imagine walking along the boundary of a park – the distance you cover is the perimeter.
Area is the amount of surface covered inside the figure. Think of how much grass grows inside the park – that’s the area.
Let us now explore these concepts for four fundamental shapes: triangle, square, rectangle, and circle.
1. Triangle
A triangle is a closed figure with three sides.
Perimeter of a triangle = sum of all three sides. If sides are $a$, $b$, $c$, then perimeter $= a + b + c.$
Area depends on the type of triangle.
For a right-angled triangle, area $= \frac{1}{2} × base × height$.
For any triangle with sides $a$, $b$, $c$, Heron’s formula can be used:
Area $= \sqrt([s(s – a)(s – b)(s – c)])$, where $s = \frac{(a + b + c)}{2}$ (the semi-perimeter).
Example: A right-angled triangle with base = $6$ cm and height = $8$ cm has area $= \frac{1}{2} × 6 × 8 = 24$ cm².
From Pythagoras Theorem, it follows that, the third side of the triangle is $10$ cm. Then perimeter = $6 + 8 + 10 = 24$ cm.
2. Square
A square has four equal sides and four right angles. Perimeter and Area can be computed as - Perimeter = 4 × side, and Area = side × side = side².
Example: A square of side $7$ m has perimeter $= 4 × 7 = 28$ m.
Area $= 7 × 7 = 49$ m².
3. Rectangle
A rectangle has opposite sides equal and all angles are right angles. The perimeter and area of a rectangle is given by - Perimeter = 2 × (length + breadth) and Area = length × breadth.
Example: A rectangle of length $12$ cm and breadth 5 cm has perimeter $= 2 × (12 + 5) = 34$ cm.
Area $= 12 × 5 = 60$ cm².
4. Circle
A circle is the set of all points at equal distance from a fixed point (centre). The distance is called the radius ($r$).
The perimeter of a circle is called the circumference and the formulas for circumference and area are as follows.
Circumference $= 2\pi r$ ($\pi ≈ 3.14$ or $22/7$) and Area $= \pi.r^2$.
Example: A circle with radius $14$ cm has circumference $= 2 × 22/7 × 14 = 88$ cm.
Area $= 22/7 × 14 × 14 = 616$ cm².
Problems
1. A square playground has side $15$ m. Find its perimeter and area.
Solution: Perimeter $= 4 × 15 = 60$ m. Area $= 15 × 15 = 225$ m².
2. A rectangular hall is $18$ m long and $12$ m wide. A carpet is to cover the floor. What area will the carpet cover?
Solution: Area $= 18 × 12 = 216$ m².
3. A triangular field has base $20$ m and height $15$ m. Find its area.
Solution: Area $= \frac{1}{2} × 20 × 15 = 150$ m².
4. A circle has radius $7$ cm. Find its circumference and area.
Solution: Circumference $= 2 × \frac{22}{7} × 7 = 44$ cm.
Area $= \frac{22}{7} × 7 × 7 = 154$ cm².
5. A farmer wants to fence his rectangular plot of length $50$ m and breadth $30$ m. How much fencing wire will he need?
Solution: Perimeter $= 2 × (50 + 30) = 160$ m.
6. A triangular park has sides $13$ m, $14$ m, and $15$ m. Find its area using Heron’s formula.
Solution: Semi-Perimeter of the triangular park, $s = \frac{(13 + 14 + 15)}{2} = 21$.
Area $= \sqrt([21(21 – 13)(21 – 14)(21 – 15)]) = \sqrt([21 × 8 × 7 × 6]) = \sqrt(7056) = 84$ m².
7. A square and a rectangle have the same perimeter of $40$ m. If the rectangle’s length is $12$ m, find its breadth and compare the areas of both shapes.
Solution: For square: Perimeter $= 40$ m ⇒ side $= 40/4 = 10$ m. Area of square = $100$ m².
For rectangle: Perimeter $= 40$ m ⇒ $2(length + breadth) = 40$ ⇒ $length + breadth = 20$ ⇒ breadth $= 20 - 12 = 8$ m.
Area $= 12 × 8 = 96$ m².
Hence, the square has a slightly larger area.
8. A circular pond has diameter $28$ m. A man walks around it once. How much distance does he cover?
Solution: Radius $= 14$ m. Circumference $= 2 × \frac{22}{7} × 14 = 88$ m.
9. A square park has side $25$ m. A path of width $2$ m runs inside along the boundary. Find the area of the path.
Solution: Outer side $= 25$ m, inner side $= 21$ m.
Area of park $= 25² = 625$ m².
Area without path $= 21² = 441$ m².
Path area $= 625 – 441 = 184$ m².
10. A semicircular garden has diameter $14$ m. Find its area.
Solution: Radius $= 7$ m. Full circle area $= πr² = \frac{22}{7} × 7 × 7 = 154$ m².
Semicircle area $= 154/2 = 77$ m².
11. A rectangular garden measures $40$ m by $30$ m. Inside it, a square flowerbed of the largest possible size is made. Find the area of the unused space.
Solution: Largest square possible will have side equal to the smaller side of rectangle, i.e., $30$ m.
Area of garden $= 40 × 30 = 1200$ m².
Area of square flowerbed $= 30² = 900$ m².
Unused space $= 1200 – 900 = 300$ m².
12. The perimeter of a triangle is $48$ cm. Two sides are $15$ cm and $17$ cm. Find the area of the triangle.
Solution: Third side $= 48 – (15 + 17) = 16$ cm.
So the sides of the triangle are of length $15$ cm, $16$ cm, $17$ cm.
Semi-perimeter $s = \frac{(15 + 16 + 17)}{2} = 24$.
Area $= \sqrt(24(24 – 15)(24 – 16)(24 – 17)) = \sqrt(24 × 9 × 8 × 7) = \sqrt(12096) = 110$ cm².
13. A circular park has a radius $21$ m. A path of width $3$ m is made around it. Find the area of the path.
Solution: Outer radius = Radius of the cirular park + Width of the path $= 21 + 3 = 24$ m.
Outer area $= \pi × 24² = \pi × 576$.
Inner area (that means, area of the circular park) $= \pi × 21² = \pi × 441$.
Therefore, area of the path $= \pi.(576 – 441) = \pi × 135 = 135 × \frac{22}{7} = 423.43$ m² (approx).
14. A square and a circle have equal areas. The side of the square is $14$ cm. Find the radius of the circle.
Solution: Area of the square $= 14 × 14 = 196$ cm².
Area of the Circle $= \pi.r^2 = 196$.
$r^2 = 196/π ≈ 196/3.14 ≈ 62.42$.
$r ≈ 7.9$ cm.
