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Math of Space: Surface Area and Volume - Areas and Volumes Around Us

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Cuboid is a three-dimensional shape with six rectangular faces. Its Surface Area is the sum of the areas of all faces, and its Volume is the product of its length, breadth, and height.

A 3D representation of a cuboid with length l, breadth b, and height h.
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A Right Circular Cylinder consists of two congruent circular bases and a curved surface. The Lateral Surface Area (2πrh2\pi rh) covers only the side, while the Total Surface Area includes the two circular ends.

Cylinder diagram showing height h and radius r.
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A Right Circular Cone is a 3D shape with a circular base and a single vertex. It has a vertical height hh, a radius rr, and a slant height ll such that l=h2+r2l = \sqrt{h^2 + r^2}.

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A Sphere is a perfectly symmetrical round object where every point on the surface is at a constant distance rr (radius) from the center. A Hemisphere is half of a sphere.

📐Formulae

Surface Area of Cuboid=2(lb+bh+hl)\text{Surface Area of Cuboid} = 2(lb + bh + hl)

Volume of Cuboid=l×b×h\text{Volume of Cuboid} = l \times b \times h

Total Surface Area of Cylinder=2πr(r+h)\text{Total Surface Area of Cylinder} = 2\pi r(r + h)

Volume of Cylinder=πr2h\text{Volume of Cylinder} = \pi r^2 h

Curved Surface Area of Cone=πrl\text{Curved Surface Area of Cone} = \pi rl

Volume of Cone=13πr2h\text{Volume of Cone} = \frac{1}{3} \pi r^2 h

Surface Area of Sphere=4πr2\text{Surface Area of Sphere} = 4\pi r^2

Volume of Sphere=43πr3\text{Volume of Sphere} = \frac{4}{3} \pi r^3

💡Examples

Problem 1:

A cylindrical water tank has a radius of 7 m7\text{ m} and a height of 10 m10\text{ m}. Calculate the cost of painting its curved surface area at the rate of Rs 2020 per m2\text{m}^2. (Use π=227\pi = \frac{22}{7})

Cylindrical tank with dimensions labeled.

Solution:

r=7 m,h=10 mr = 7\text{ m}, h = 10\text{ m} Curved Surface Area (CSA)=2πrh\text{Curved Surface Area (CSA)} = 2\pi rh CSA=2×227×7×10\text{CSA} = 2 \times \frac{22}{7} \times 7 \times 10 CSA=440 m2\text{CSA} = 440\text{ m}^2 Total Cost=440×20=8800\text{Total Cost} = 440 \times 20 = 8800 Total Cost=Rs 8800\text{Total Cost} = \text{Rs } 8800

Explanation:

To find the painting cost, we first calculate the Curved Surface Area (CSA) of the cylinder using the formula 2πrh2\pi rh. Then, we multiply the area by the unit cost of painting.

Problem 2:

Find the volume of a sphere whose surface area is 154 cm2154 \text{ cm}^2. (Use π=227\pi = \frac{22}{7})

Sphere with radius r indicated.

Solution:

Surface Area=4πr2=154\text{Surface Area} = 4\pi r^2 = 154 4×227×r2=1544 \times \frac{22}{7} \times r^2 = 154 r2=154×74×22r^2 = \frac{154 \times 7}{4 \times 22} r2=7×74=494r^2 = \frac{7 \times 7}{4} = \frac{49}{4} r=72=3.5 cmr = \frac{7}{2} = 3.5 \text{ cm} Volume=43πr3=43×227×72×72×72\text{Volume} = \frac{4}{3} \pi r^3 = \frac{4}{3} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times \frac{7}{2} Volume=11×493=5393≈179.67 cm3\text{Volume} = \frac{11 \times 49}{3} = \frac{539}{3} \approx 179.67 \text{ cm}^3

Explanation:

First, use the given surface area to find the radius rr of the sphere. Once rr is known, substitute it into the volume formula V=43πr3V = \frac{4}{3}\pi r^3.