The inner and outer diameters of ring 1 of a dartboard are 15 cm and 17 cm respectively and those of ring II are 12 cm and 13 cm respectively. What is the total area of these two rings?
To find the total area of the two rings, we will first calculate the area of each ring separately and then add them up.
For Ring I:
- Outer diameter of Ring I = 17 cm, so the outer radius \(R_1 = \frac{17}{2} = 8.5\) cm.
- Inner diameter of Ring I = 15 cm, so the inner radius \(r_1 = \frac{15}{2} = 7.5\) cm.
- Area of Ring I = Area of outer circle - Area of inner circle
- The area of Ring I = \(\pi R_1^2 - \pi r_1^2 = \pi (8.5^2 - 7.5^2)\)
- \(= \pi (72.25 - 56.25) = \pi (16) = 16\pi \, \text{cm}^2\)
For Ring II:
- Outer diameter of Ring II = 13 cm, so the outer radius \(R_2 = \frac{13}{2} = 6.5\) cm.
- Inner diameter of Ring II = 12 cm, so the inner radius \(r_2 = \frac{12}{2} = 6\) cm.
- Area of Ring II = Area of outer circle - Area of inner circle
- The area of Ring II = \(\pi R_2^2 - \pi r_2^2 = \pi (6.5^2 - 6^2)\)
- \(= \pi (42.25 - 36) = \pi (6.25) = 6.25\pi \, \text{cm}^2\)
Adding the areas of both rings:
- Total area = Area of Ring I + Area of Ring II
- \(= 16\pi + 6.25\pi = 22.25\pi = 89\pi \, \text{cm}^2\)
Hence, the total area of the two rings is \(89\pi \text{ cm}^2\).
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Important Questions from 2-D Mensuration
- The length of a rectangular plot is $(x^2 + xy + y^2)$ m and its breadth is $(x^2 - 5xy - y^2)$ m.
Find its perimeter, when $x = 1$ and $y = -1$. - The ratio between the perimeter and breadth of a rectangle is 3: 1. If the area of the rectangle is $98 \text{ cm}^2$, find the perimeter (in cm) of the rectangle.
The shorter side of a rectangle is 15 cm less than the longer side. The numerical value of its area is equal to 5 times the numerical value of its perimeter. What is the length (in cm) of its longer side?
- In a circle of radius 10.5 cm, if the angle of a sector is $\frac{2\pi}{3}$, then the perimeter of the sector is (in cm):
(Take $\pi = \frac{22}{7}$) - Find the circumference (in m) of the largest circle that can be inscribed in a rectangle whose dimensions are given as 114 m and 63 m.
Take $\pi = \frac{22}{7}$
