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Examples 6 to 10 of Mensuration

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(6) Find the area of an isosceles triangle, in which each one of equal sides is 7 m and the third side 8 m.

Soln. Here a = 7, b = 7 and c = 8.

∴ s = $\frac{a+b+c}{2}$ = 10

∴ Area = $\sqrt{s\ (s\ -\ a\ )\ (\ s\ -\ b\ )\ (\ s\ -\ c\ )}$

= $\sqrt{10\times3\times3\times2}$

= $6\sqrt5$ m².

(7) Find the area of an equilateral triangle, whose side measures 10 cm. each.

Soln. Area of the triangle = $\frac{{\sqrt3} x^2}{4}$

= $\frac{\sqrt3}{4} \times 10 \times 10 = 25\sqrt3$ cm².

(8) Find the area of regular hexagon whose one side is 4 m.

Soln. Hexagon consists of 6 equilateral triangles.

∴ Its area = 6 × Area of each equilateral triangle = 6 × $\frac{\sqrt3}{4}$ ( side )²

= $\frac{6\sqrt3}{4}$ × 4² = 24 $\sqrt3$ m².

(9) A rope 88 cm long has been put in the form of a circle. Find the area of the circle so formed.

Soln. Here, Circumference of the circle = 88 cm.

∴ 2π r = 88 ⇒ r = 14 cm

Now, Area of the circle = π r² = 616 cm².

(10) The area of ring is 1232 cm². If the radius of the smaller circle is 7 cm, find the radius of the bigger circle.

Soln. Area of the ring = ( R² – r² )

; Where, R is the outer radius and r is the inner radius.

∴1232 = $\frac{22}{7}$ ( R² – 7²) ⇒ R² – 49 = $\frac{1232\times7}{22}$ = 392

⇒ R² = 392 + 49 = 441 ⇒ R = 21 cm.