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Mensuration - Volume and Surface Area of Prisms and Cylinders

Grade 8IGCSE

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

🔑Concepts

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Definition of a Prism: A 3D solid with a constant cross-section throughout its length.

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Volume: The amount of 3D space an object occupies, measured in cubic units (cm3cm^3, m3m^3).

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Total Surface Area: The sum of the areas of all the faces (outer surfaces) of a 3D solid.

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Uniform Cross-section: The shape remains the same when sliced parallel to the base (e.g., the circle in a cylinder or the triangle in a triangular prism).

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Cylinders: Though curved, cylinders are treated as circular prisms where the cross-section is a circle.

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Unit Conversion: Remember that 1cm3=1000mm31 cm^3 = 1000 mm^3 and 1m3=1,000,000cm31 m^3 = 1,000,000 cm^3.

📐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.