Mensuration — Maharashtra (SSC) Class 10 Mathematics Solutions (Free)
Free step-by-step Maharashtra (SSC) Class 10 Mathematics solutions for "Mensuration" — important questions with detailed answers, download PDF for board exam preparation.
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Mensuration — Textbook Questions Answered (8)
Q1: Find the curved surface area of a cone with base radius 7 cm and slant height 10 cm.
Step 1: Curved surface area (CSA) of cone = π * r * l, where r is radius and l is slant height.
Step 2: CSA = π * 7 * 10 = 70π cm^2.
Step 3: If numerical value needed, CSA = 70 * 22/7 = 10 * 22 = 220 cm^2 (using π ≈ 22/7).
Answer: Curved surface area = 70π cm^2 or 220 cm^2.
Q2: A sphere has radius 6 cm. Find its surface area and volume.
Step 1: Surface area of sphere = 4πr^2, where r = radius.
Step 2: Surface area = 4π(6)^2 = 4π * 36 = 144π cm^2.
Step 3: Using π ≈ 22/7: Surface area = 144 * 22/7 = 3168/7 ≈ 452.57 cm^2.
Step 4: Volume of sphere = (4/3)πr^3.
Step 5: Volume = (4/3)π(6)^3 = (4/3)π * 216 = 288π cm^3.
Step 6: Using π ≈ 22/7: Volume = 288 * 22/7 = 6336/7 ≈ 905.14 cm^3.
Answer: Surface area = 144π cm^2 (≈ 452.57 cm^2), Volume = 288π cm^3 (≈ 905.14 cm^3).
Q3: A cylinder has radius 5 cm and height 12 cm. Find total surface area.
Step 1: Total surface area of cylinder = 2πr^2 + 2πrh = 2πr(r + h), where r is radius and h is height.
Step 2: Total SA = 2π(5)(5 + 12) = 2π(5)(17) = 170π cm^2.
Step 3: Using π ≈ 22/7: Total SA = 170 * 22/7 = 3740/7 ≈ 534.29 cm^2.
Alternatively: Two circular bases = 2πr^2 = 2π(25) = 50π cm^2.
Curved surface area = 2πrh = 2π(5)(12) = 120π cm^2.
Total = 50π + 120π = 170π cm^2.
Answer: Total surface area = 170π cm^2 or approximately 534.29 cm^2.
Q4: A sector of a circle with radius 10 cm subtends an angle of 36 degrees at the center. Find its area.
Step 1: Area of sector = (θ/360°) * πr^2, where θ is central angle in degrees and r is radius.
Step 2: Area = (36/360) * π(10)^2 = (1/10) * 100π = 10π cm^2.
Step 3: Using π ≈ 22/7: Area = 10 * 22/7 = 220/7 ≈ 31.43 cm^2.
Answer: Area of sector = 10π cm^2 or approximately 31.43 cm^2.
Q5: Find the volume of a rectangular box with length 8 cm, width 6 cm, and height 5 cm.
Step 1: Volume of rectangular box = length * width * height.
Step 2: Volume = 8 * 6 * 5 = 240 cm^3.
Answer: Volume = 240 cm^3.
Q6: A cone has base radius 4 cm and height 6 cm. Find its volume and total surface area.
Step 1: Volume of cone = (1/3)πr^2h.
Step 2: Volume = (1/3)π(4)^2(6) = (1/3)π * 16 * 6 = 32π cm^3.
Step 3: Using π ≈ 22/7: Volume = 32 * 22/7 = 704/7 ≈ 100.57 cm^3.
Step 4: For surface area, first find slant height l using l^2 = r^2 + h^2.
Step 5: l^2 = 4^2 + 6^2 = 16 + 36 = 52. So l = sqrt(52) = 2*sqrt(13) cm.
Step 6: Total surface area = πr^2 + πrl = πr(r + l) = π(4)(4 + 2*sqrt(13)).
Step 7: Total SA = 4π(4 + 2*sqrt(13)) = 16π + 8π*sqrt(13) cm^2.
Answer: Volume = 32π cm^3 (≈ 100.57 cm^3), Total surface area = (16π + 8π*sqrt(13)) cm^2.
Q7: A hemisphere has radius 8 cm. Find its curved surface area and volume.
Step 1: Curved surface area of hemisphere = 2πr^2 (half of sphere's surface area).
Step 2: CSA = 2π(8)^2 = 2π * 64 = 128π cm^2.
Step 3: Using π ≈ 22/7: CSA = 128 * 22/7 = 2816/7 ≈ 402.29 cm^2.
Step 4: Volume of hemisphere = (2/3)πr^3 (half of sphere's volume).
Step 5: Volume = (2/3)π(8)^3 = (2/3)π * 512 = (1024/3)π cm^3.
Step 6: Using π ≈ 22/7: Volume = (1024/3) * 22/7 = 22528/21 ≈ 1072.76 cm^3.
Answer: Curved surface area = 128π cm^2 (≈ 402.29 cm^2), Volume = (1024/3)π cm^3 (≈ 1072.76 cm^3).
Q8: A square pyramid has base edge 5 cm and height 12 cm. Find its volume and lateral surface area.
Step 1: Volume of pyramid = (1/3) * base area * height.
Step 2: Base area = 5^2 = 25 cm^2.
Step 3: Volume = (1/3) * 25 * 12 = 100 cm^3.
Step 4: For lateral surface area, find slant height. The slant height l from apex to midpoint of base edge.
Step 5: Distance from center to midpoint of edge = 5/2 = 2.5 cm.
Step 6: l^2 = 12^2 + 2.5^2 = 144 + 6.25 = 150.25. So l = sqrt(150.25) ≈ 12.26 cm.
Step 7: Lateral surface area = (1/2) * perimeter of base * slant height = (1/2) * (4 * 5) * 12.26 = (1/2) * 20 * 12.26 = 122.6 cm^2.
Alternatively, each triangular face has area = (1/2) * 5 * l. Four faces: 4 * (1/2) * 5 * l = 10l cm^2.
Answer: Volume = 100 cm^3, Lateral surface area ≈ 122.6 cm^2.
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