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Mensuration - Volume and Surface Area of Prisms and Cylinders

Grade 8Cambridge (IGCSE)

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

🔑Concepts

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A prism is a 3D shape with a constant cross-section throughout its length. The volume is always calculated by multiplying the area of this cross-section by the length or height of the prism.

A triangular prism illustrating the constant cross-section area and the length.
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A cylinder is a special type of prism where the cross-section is a circle. The volume is given by V=πr2hV = \pi r^2 h.

Cylinder with radius r and height h.
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The Total Surface Area (TSA) of a prism is the sum of the areas of all its faces. This includes two identical base areas and the lateral area (perimeter of base ×\times length).

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The curved surface of a cylinder, when 'unrolled', forms a rectangle where the length is the circumference of the circle (2πr2 \pi r) and the width is the height (hh).

📐Formulae

Volume of any Prism=Area of Cross-section×Length\text{Volume of any Prism} = \text{Area of Cross-section} \times \text{Length}

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

Total Surface Area of a Cuboid=2(lw+lh+wh)\text{Total Surface Area of a Cuboid} = 2(lw + lh + wh)

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

Curved Surface Area of a Cylinder=2πrh\text{Curved Surface Area of a Cylinder} = 2 \pi r h

Total Surface Area of a Cylinder=2πr2+2πrh\text{Total Surface Area of a Cylinder} = 2 \pi r^2 + 2 \pi r h

Surface Area of a Prism=(2×Base Area)+(Perimeter of Base×Length)\text{Surface Area of a Prism} = (2 \times \text{Base Area}) + (\text{Perimeter of Base} \times \text{Length})

💡Examples

Problem 1:

A triangular prism has a right-angled triangular base with sides 3 cm, 4 cm, and 5 cm. The length of the prism is 10 cm. Calculate its Volume.

Solution:

Volume=60cm3Volume = 60 cm^3

Explanation:

  1. Find the area of the triangular cross-section: Area=12×base×height=12×3×4=6cm2Area = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 4 = 6 cm^2. 2. Multiply by the length: Volume=6cm2×10cm=60cm3Volume = 6 cm^2 \times 10 cm = 60 cm^3.

Problem 2:

Calculate the total surface area of a cylinder with a radius of 7 cm and a height of 10 cm. (Use π=227\pi = \frac{22}{7})

Solution:

SA=748cm2SA = 748 cm^2

Explanation:

  1. Area of two circular bases: 2×πr2=2×227×72=2×154=308cm22 \times \pi r^2 = 2 \times \frac{22}{7} \times 7^2 = 2 \times 154 = 308 cm^2. 2. Curved surface area: 2πrh=2×227×7×10=440cm22 \pi r h = 2 \times \frac{22}{7} \times 7 \times 10 = 440 cm^2. 3. Total Surface Area = 308+440=748cm2308 + 440 = 748 cm^2.

Problem 3:

A rectangular water tank (cuboid) measures 2m by 1.5m by 1m. How many liters of water can it hold?

Solution:

3000 Liters3000 \text{ Liters}

Explanation:

  1. Calculate volume in m3m^3: V=2×1.5×1=3m3V = 2 \times 1.5 \times 1 = 3 m^3. 2. Convert to liters: Since 1m3=10001 m^3 = 1000 liters, 3×1000=30003 \times 1000 = 3000 liters.

Problem 4:

A cylindrical pipe has an internal radius of 4 cm4 \text{ cm} and a length of 25 cm25 \text{ cm}. Calculate the volume of water the pipe can hold in terms of π\pi.

A horizontal cylindrical pipe with radius 4cm and length 25cm.

Solution:

Volume=πr2h\text{Volume} = \pi r^2 h Volume=π×42×25\text{Volume} = \pi \times 4^2 \times 25 Volume=π×16×25\text{Volume} = \pi \times 16 \times 25 Volume=400π cm3\text{Volume} = 400\pi \text{ cm}^3

Explanation:

To find the volume, identify the radius r=4 cmr = 4 \text{ cm} and the height (length) h=25 cmh = 25 \text{ cm}. Substitute these values into the cylinder volume formula.

Problem 5:

Find the total surface area of a L-shaped prism with a constant cross-sectional area of 20 cm220 \text{ cm}^2, a base perimeter of 24 cm24 \text{ cm}, and a length of 15 cm15 \text{ cm}.

An L-shaped prism showing the base area and length.

Solution:

TSA=(2×Base Area)+(Perimeter×Length)\text{TSA} = (2 \times \text{Base Area}) + (\text{Perimeter} \times \text{Length}) TSA=(2×20)+(24×15)\text{TSA} = (2 \times 20) + (24 \times 15) TSA=40+360\text{TSA} = 40 + 360 TSA=400 cm2\text{TSA} = 400 \text{ cm}^2

Explanation:

The total surface area of any prism is the sum of the areas of the two identical bases and the area of the rectangular sides (lateral area). The lateral area is the perimeter of the base multiplied by the prism's length.