CBSE Class 10 Maths: Areas Related to Circles Ex 12.3

Understanding Areas of Combined Plane Figures

In previous exercises, you learned how to calculate areas of basic geometric shapes like circles, sectors, and segments. Now, in Exercise 12.3, the challenge is to find the area of regions that are formed by combining two or more of these basic figures. Often, these problems present a 'shaded region' whose area needs to be determined.

The key to solving these problems lies in a systematic approach:

1. Decomposition: Break down the complex shaded region into simpler, recognizable geometric shapes. For example, a shaded region might be a square with a circle removed, or a sector of a circle with a triangle removed.
2. Identification of Dimensions: Carefully identify the radii, side lengths, angles, bases, and heights of all the component shapes. These might be directly given or require calculation using properties of geometry (e.g., Pythagoras theorem, properties of squares or equilateral triangles).
3. Application of Formulas: Apply the appropriate area formulas for each identified simple shape:

• Area of Circle = πr²
• Area of Sector = (θ/360°) × πr²
• Area of Segment = Area of Sector - Area of corresponding Triangle
• Area of Square = side²
• Area of Rectangle = length × breadth
• Area of Triangle = ½ × base × height
• Area of Equilateral Triangle = (√3/4) × side²

1. Combination/Subtraction: Add or subtract the areas of the component shapes as required to find the area of the desired shaded region. Sometimes, you might need to find the area of a larger figure and subtract the areas of the unshaded parts.

Remember to always keep track of units and use the value of π as specified in the problem (usually 22/7 or 3.14). A clear diagram and labeling will greatly assist in visualizing the problem and preventing errors.

Step-by-Step Strategy for Solving Combined Figure Problems

1. Step 1: Analyze the Diagram and Identify Shapes — Carefully observe the given figure. What basic geometric shapes (circles, squares, triangles, sectors, etc.) can you identify? Is the shaded region a combination (sum) or a leftover (difference) of these shapes? Sketch the figure and label all known dimensions.
2. Step 2: List Known and Unknown Dimensions — Write down all the given measurements like radii, side lengths, angles. If some dimensions are not directly given, think about how you can calculate them using other information or geometric properties (e.g., diameter = 2 × radius, diagonals of a square, properties of tangents).
3. Step 3: Formulate a Plan to Find the Shaded Area — Decide on the exact mathematical operations. Will you add areas? Subtract areas? For example, 'Area of Shaded Region = Area of Square - Area of Four Quadrants' or 'Area of Shaded Region = Area of Sector + Area of Triangle'.
4. Step 4: Calculate Individual Areas — Compute the area of each component shape identified in Step 1 using the appropriate formulas and dimensions from Step 2. Be mindful of using the correct value for π (e.g., 22/7 or 3.14) and maintaining accuracy in calculations.
5. Step 5: Perform Final Calculation and State Units — Combine or subtract the individual areas as per your plan from Step 3. Ensure the final answer is accompanied by the correct units (e.g., cm², m²). Double-check your calculations, especially for arithmetic errors.

Worked Examples

• Example 1: Finding the area of a shaded region in a square. The figure shows a square OABC inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14) Solution: Step 1: Understand the figure. The shaded region is formed by subtracting the area of the square OABC from the area of the quadrant OPBQ. To find the area of the quadrant, we need its radius. The radius of the quadrant (r) is the length of OB. OA = 20 cm, so the side of the square is 20 cm. Step 2: Calculate the radius of the quadrant. In right-angled triangle OAB, by Pythagoras theorem: OB² = OA² + AB² OB² = 20² + 20² OB² = 400 + 400 OB² = 800 OB = √800 = 20√2 cm. So, r = 20√2 cm. Step 3: Calculate the area of the quadrant OPBQ. Area of quadrant = (θ/360°) × πr² = (90°/360°) × 3.14 × (20√2)² = (1/4) × 3.14 × (400 × 2) = (1/4) × 3.14 × 800 = 3.14 × 200 = 628 cm². Step 4: Calculate the area of the square OABC. Area of square = side² = 20² = 400 cm². Step 5: Calculate the area of the shaded region. Area of shaded region = Area of quadrant - Area of square = 628 cm² - 400 cm² = 228 cm². Final Answer: The area of the shaded region is 228 cm².
• Example 2: Area of a shaded region formed by two arcs. Two circular arcs have their centers at point O. The radius of the inner arc is 7 cm and the outer arc is 14 cm. If ∠AOC = 40°, find the area of the shaded region formed between the arcs. (Use π = 22/7) Solution: Step 1: Understand the figure. The shaded region is the area between two concentric sectors, OAC (larger) and OBD (smaller). The area of the shaded region is the difference between the area of the larger sector and the smaller sector. Step 2: Identify radii and angle. Radius of larger sector (R) = 14 cm (OA = OC) Radius of smaller sector (r) = 7 cm (OB = OD) Angle of both sectors (θ) = 40°. Step 3: Calculate the area of the larger sector OAC. Area of sector OAC = (θ/360°) × πR² = (40°/360°) × (22/7) × 14² = (1/9) × (22/7) × 196 = (1/9) × 22 × 28 = 616/9 cm². Step 4: Calculate the area of the smaller sector OBD. Area of sector OBD = (θ/360°) × πr² = (40°/360°) × (22/7) × 7² = (1/9) × (22/7) × 49 = (1/9) × 22 × 7 = 154/9 cm². Step 5: Calculate the area of the shaded region. Area of shaded region = Area of sector OAC - Area of sector OBD = 616/9 cm² - 154/9 cm² = (616 - 154)/9 cm² = 462/9 cm² = 154/3 cm². Final Answer: The area of the shaded region is 154/3 cm² (approximately 51.33 cm²).

YoLearn AI Tutor's Exam Tips for Exercise 12.3

1. Draw and Label Carefully: Always redraw the figure and clearly label all given dimensions. If auxiliary lines are needed (e.g., to form a right triangle), draw them. This visual aid reduces confusion and helps in planning your solution.

2. Choose the Correct Formula for π: Pay close attention to the value of π specified in the question (22/7 or 3.14). Using the wrong value will lead to incorrect answers, even if your method is correct.

3. Units, Units, Units: Always include the correct units (cm², m², etc.) with your final answer. Missing units can lead to deduction of marks.

4. Break Down Complex Shapes: The most common mistake is trying to find the area of the shaded region directly. Instead, break it down: identify larger shapes, smaller shapes, and then decide whether to add or subtract their areas.

5. Double-Check Calculations: Area problems often involve multiple steps and arithmetic. After you've found an answer, quickly review each calculation step to catch any silly mistakes.

6. Practice with a Variety of Figures: This exercise features squares, triangles, and sectors combined in many ways. Practice different configurations to become adept at identifying component shapes quickly.

Practice Questions with Solutions

• Q: A square ABCD has a side of 14 cm. Four congruent circles are drawn such that each circle touches two adjacent circles and two sides of the square. Find the area of the shaded region (the area of the square not covered by the circles). (Use π = 22/7) A: Step 1: Determine the radius of each circle. Since four congruent circles touch each other and the sides of the square, the diameter of each circle is half the side of the square. Diameter = 14 cm / 2 = 7 cm. Radius (r) = 7 cm / 2 = 3.5 cm. Step 2: Calculate the area of the square. Area of square = side² = 14² = 196 cm². Step 3: Calculate the area of one circle. Area of one circle = πr² = (22/7) × (3.5)² = (22/7) × 12.25 = 22 × 1.75 = 38.5 cm². Step 4: Calculate the total area of the four circles. Total area of 4 circles = 4 × 38.5 cm² = 154 cm². Step 5: Calculate the area of the shaded region. Area of shaded region = Area of square - Total area of 4 circles = 196 cm² - 154 cm² = 42 cm². Final answer: The area of the shaded region is 42 cm².
• Q: In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in the figure. Find the area of the design. (Use π = 22/7 and √3 = 1.732) A: Step 1: Understand the figure and identify the shapes. The design is the area of the circular table cover minus the area of the equilateral triangle ABC. The center of the circle is the circumcenter of the equilateral triangle. Let R be the radius of the circle = 32 cm. Step 2: Find the side of the equilateral triangle. For an equilateral triangle inscribed in a circle, the relation between the side (a) and the circumradius (R) is R = a/√3. So, a = R√3 = 32√3 cm. Step 3: Calculate the area of the equilateral triangle. Area of equilateral triangle = (√3/4) × a² = (√3/4) × (32√3)² = (√3/4) × (1024 × 3) = √3 × 256 × 3 = 768√3 cm². Using √3 = 1.732, Area = 768 × 1.732 = 1330.176 cm². Step 4: Calculate the area of the circular table cover. Area of circle = πR² = (22/7) × 32² = (22/7) × 1024 = 22528/7 cm² ≈ 3218.28 cm². Step 5: Calculate the area of the design (shaded region). Area of design = Area of circle - Area of equilateral triangle = (22528/7) - 768√3 cm² (Exact answer) = 3218.28 - 1330.176 = 1888.104 cm² (Approximate answer). Final answer: The area of the design is approximately 1888.10 cm².
• Q: From each corner of a square of side 4 cm, a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut from the centre. Find the area of the remaining portion of the square. (Use π = 22/7) A: Step 1: Identify the shapes and dimensions. Area of square = side² = 4² = 16 cm². There are 4 quadrants, each with radius r = 1 cm. There is 1 central circle with diameter 2 cm, so its radius R = 1 cm. Step 2: Calculate the area of the four quadrants. Area of one quadrant = (1/4)πr² = (1/4) × (22/7) × 1² = 22/28 = 11/14 cm². Area of four quadrants = 4 × (11/14) = 44/14 = 22/7 cm². Step 3: Calculate the area of the central circle. Area of central circle = πR² = (22/7) × 1² = 22/7 cm². Step 4: Calculate the area of the removed portions. Total area removed = Area of 4 quadrants + Area of central circle = (22/7) + (22/7) = 44/7 cm². Step 5: Calculate the area of the remaining portion. Area of remaining portion = Area of square - Total area removed = 16 - (44/7) = (112 - 44)/7 = 68/7 cm². Final answer: The area of the remaining portion of the square is 68/7 cm².
• Q: A round table cover has six equal designs as shown in the figure. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹0.35 per cm². (Use √3 = 1.7) A: Step 1: Understand the figure and calculate the angle of each segment. The table cover has six equal designs. These designs are segments of the circle. The center of the circle forms 6 equal sectors, so the angle of each sector (θ) = 360°/6 = 60°. The radius (r) = 28 cm. Step 2: Determine the type of triangle formed by the radii. Since the angle of each sector is 60° and the two sides (radii) are equal, the triangle formed by the two radii and the chord (e.g., ΔOAB) is an equilateral triangle. So, OA = OB = AB = 28 cm. Step 3: Calculate the area of one sector and one triangle. Area of one sector = (θ/360°) × πr² = (60°/360°) × (22/7) × 28² = (1/6) × (22/7) × 784 = (1/6) × 22 × 112 = 2464/6 = 1232/3 cm². Area of equilateral triangle = (√3/4) × side² = (1.7/4) × 28² = (1.7/4) × 784 = 1.7 × 196 = 333.2 cm². Step 4: Calculate the area of one design (segment). Area of one design = Area of sector - Area of triangle = (1232/3) - 333.2 cm² = 410.67 - 333.2 = 77.47 cm² (approx). Step 5: Calculate the total area of the six designs. Total area of designs = 6 × 77.47 cm² = 464.82 cm² (approx). Step 6: Calculate the cost of making the designs. Cost per cm² = ₹0.35. Total cost = Total area × Rate = 464.82 × 0.35 = ₹162.687. Final answer: The cost of making the designs is approximately ₹162.69.

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