Area and Perimeter Formulas Reference

Rectangle and square

\text{Rectangle: Area} = l \times b, \qquad \text{Perimeter} = 2(l+b)

\text{Square: Area} = a^2, \qquad \text{Perimeter} = 4a

Circle and sector

\text{Circle: Area} = \pi r^2, \qquad \text{Circumference} = 2\pi r

A sector is the pie-slice region between two radii and an arc, spanning an angle $C$ in degrees:

\text{Area of sector} = \pi r^2 \times \dfrac{C}{360}, \qquad \text{Perimeter of sector} = 2r + 2\pi r \times \dfrac{C}{360}

Worked example: a sector

A garden bed is a quarter circle of radius 14 m. Find its area and the length of edging needed around it.

Solution: With $C = 90$ , the area is $\pi(14)^2 \times \dfrac{90}{360} = 49\pi \approx 153.94$ m². The edging runs along both straight radii and the arc:

2(14) + 2\pi(14) \times \dfrac{90}{360} = 28 + 7\pi \approx 49.99 \text{ m}

Triangle, including Heron's formula

With a known base and height, the area is $\dfrac{1}{2} \times \text{base} \times \text{height}$ . When only the three sides are known, Heron's formula avoids needing the height:

s = \dfrac{a+b+c}{2}, \qquad \text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Worked example: for sides 5, 6 and 7, $s = 9$ and the area is $\sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} = 6\sqrt{6} \approx 14.70$ .

Regular pentagon

A regular pentagon splits into 5 identical triangles meeting at its centre, so its area is 5 times one of them. With side $a$ :

\text{Area} = \dfrac{5a^2}{4\tan 36^\circ}, \qquad \text{Perimeter} = 5a

For $a = 2$ , the area is $\dfrac{20}{4\tan 36^\circ} \approx 6.88$ .

Worked example: same perimeter, different area

A rectangle is 12 cm by 8 cm. A square has the same perimeter. Which has the larger area?

Solution: The rectangle's perimeter is $2(12 + 8) = 40$ cm, so the square's side is $40 \div 4 = 10$ cm. The square's area is $100$ cm², larger than the rectangle's $96$ cm². For a fixed perimeter, the square has the largest area of any rectangle.

Common mistakes

Using the diameter as the radius. Halve it first.
Leaving out the radii in a sector's perimeter. The boundary is the arc plus two straight edges.
Mixing degrees and radians. The sector formulas above use $C$ in degrees; in radians the area is $\tfrac{1}{2}r^2\theta$ .
Using Heron's formula with the perimeter instead of the semi-perimeter. $s$ is half of $a + b + c$ .

Where this comes up in exams

The 2025-26 CBSE Class IX syllabus lists the area of a triangle using Heron's formula. Mensuration and geometry are also part of the General Aptitude syllabus in every GATE 2027 paper.

Checked against CBSE Mathematics curriculum, Classes IX-X (2025-26) and GATE 2027 information brochure (General Aptitude syllabus). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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