Area and Perimeter Class 9: notes and important questions

Area and Perimeter: revision notes

1. Perimeter and arc length

• Perimeter of a polygon: the sum of its sides. Circumference of a circle: \(C = 2\pi r = \pi d\).
• An arc of a circle that subtends \(\theta^\circ\) at the centre has length \(l = \dfrac{\theta}{360} \times 2\pi r\). The perimeter of the sector is \(l + 2r\).
• A semicircle of radius \(r\) has perimeter \(\pi r + 2r\).

2. Areas of standard shapes

Triangle \(\dfrac12 \times \text{base} \times \text{height}\); rectangle \(lb\); parallelogram \(bh\); trapezium \(\dfrac12(a + b)h\); rhombus \(\dfrac12 d_1d_2\); equilateral triangle of side \(a\): \(\dfrac{\sqrt3}{4}a^2\). Circle \(\pi r^2\); sector of angle \(\theta^\circ\): \(\dfrac{\theta}{360}\pi r^2\).

3. Heron's formula

For a triangle with sides \(a, b, c\) and semi-perimeter \(s = \dfrac{a + b + c}{2}\):

\[\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}\]

It needs only the three sides, so it works when no height is known. A quadrilateral with a known diagonal splits into two triangles.

4. Incircle

The incircle touches all three sides; its centre is where the angle bisectors meet. Joining the incentre to the vertices splits the triangle into three triangles of height \(r\), so \(\text{Area} = rs\) and \(r = \dfrac{\text{Area}}{s}\).

5. Brahmagupta's formula

For a cyclic quadrilateral with sides \(a, b, c, d\) and \(s = \dfrac{a + b + c + d}{2}\): \(\text{Area} = \sqrt{(s - a)(s - b)(s - c)(s - d)}\). Heron's formula is the case \(d = 0\).

Worked example 1

Find the area of a triangle with sides \(13\), \(14\) and \(15\) cm.

\(s = 21\); area \(= \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84\) cm².

Worked example 2

Find the length of an arc of angle \(90^\circ\) in a circle of radius \(14\) cm (take \(\pi = \dfrac{22}{7}\)).

\(l = \dfrac14 \times 2 \times \dfrac{22}{7} \times 14 = 22\) cm.

Worked example 3

Find the inradius of the triangle with sides \(6\), \(8\) and \(10\) cm.

Area \(= 24\) cm², \(s = 12\), so \(r = 2\) cm.

Common errors

• Using the perimeter instead of the semi-perimeter \(s\) in Heron's formula.
• Forgetting the two radii in the perimeter of a sector or the diameter in the perimeter of a semicircle.
• Using Brahmagupta's formula for a quadrilateral that is not cyclic.
• Mixing units (cm and m) or writing an area in cm instead of cm².

Exam tips

• Write \(s\), then \(s - a\), \(s - b\), \(s - c\) on separate lines before multiplying; factorise under the root instead of multiplying out.
• Use the value of \(\pi\) the question gives; otherwise leave the answer in terms of \(\pi\).