CIE IGCSE Mathematics (0580) Surface area and volume Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Surface area and volume Study Notes
LEARNING OBJECTIVE
- Surface area and volume of Shapes
Key Concepts:
- Surface area and volume of known shapes
Cuboid
Cuboid
A cuboid is a three-dimensional shape with six rectangular faces, twelve edges, and eight vertices. Opposite faces are equal and all angles are right angles.
Surface Area
- Total Surface Area (TSA): \( 2(lw + wh + lh) \)
- Lateral Surface Area (LSA): \( 2h(l + w) \)
Volume
\( \text{Volume} = l \times w \times h \)
Example :
A cuboid has dimensions $\mathrm{10 ~cm × 4 ~cm × 3 ~cm}$. Find its surface area and volume.
▶️ Answer/Explanation
Total Surface Area = \( 2(10 \times 4 + 4 \times 3 + 10 \times 3) = 2(40 + 12 + 30) = 2 \times 82 = 164 \ \text{cm}^2 \)
Volume = \( 10 \times 4 \times 3 = 120 \ \text{cm}^3 \)
Example :
A tank is shaped like a cuboid with dimensions $\mathrm{1.2~ m \times 80~ cm \times 0.5 ~m}$. Find the total surface area and volume in standard units.
▶️ Answer/Explanation
Convert 80 cm to meters: \( 80 \text{ cm} = 0.8 \text{ m} \)
Total Surface Area = \( 2(1.2 \times 0.8 + 0.8 \times 0.5 + 1.2 \times 0.5) = 2(0.96 + 0.4 + 0.6) = 2 \times 1.96 = 3.92 \ \text{m}^2 \)
Volume = \( 1.2 \times 0.8 \times 0.5 = 0.48 \ \text{m}^3 \)
Prism
Prism
AA prism is a solid with a uniform cross-section – that is, the same shape repeated along its length. Common examples include triangular prisms and rectangular prisms. Cylinders are also special types of prisms (i.e. with circular cross-sections).
Surface Area
- Total Surface Area (TSA): \( \text{Lateral Surface Area} + 2 \times \text{Area of Base} \)
- Lateral Surface Area (LSA): \( \text{Perimeter of base} \times \text{height} \)
Volume
\( \text{Volume} = \text{Area of cross-section} \times \text{height} \)
Example :
A triangular prism has a triangle base of area 24 cm² and height 10 cm. Find its volume.
▶️ Answer/Explanation
Volume = \( 24 \times 10 = 240 \ \text{cm}^3 \)
Example:
A prism has a trapezium cross-section with parallel sides 6 cm and 10 cm, height 4 cm. The length of the prism is 12 cm. Find the volume and total surface area if the non-parallel sides are 5 cm and 5 cm.
▶️ Answer/Explanation
Area of cross-section = \( \frac{1}{2} \times (6 + 10) \times 4 = \frac{1}{2} \times 16 \times 4 = 32 \ \text{cm}^2 \)
Volume = \( 32 \times 12 = 384 \ \text{cm}^3 \)
Perimeter of base = \( 6 + 10 + 5 + 5 = 26 \ \text{cm} \)
Lateral Surface Area = \( 26 \times 12 = 312 \ \text{cm}^2 \)
Total Surface Area = \( 312 + 2 \times 32 = 312 + 64 = 376 \ \text{cm}^2 \)
Cylinder
Cylinder
A cylinder is a 3D shape with two parallel circular bases connected by a curved surface. It has constant circular cross-section throughout its height.
Surface Area
- Curved Surface Area (CSA): \( 2 \pi r h \)
- Total Surface Area (TSA): \( 2 \pi r h + 2 \pi r^2 \)
Volume
\( \text{Volume} = \pi r^2 h \)
Example:
A cylinder has a radius of 7 cm and a height of 10 cm. Find its curved surface area and volume. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
CSA = \( 2 \times 3.14 \times 7 \times 10 = 439.6 \ \text{cm}^2 \)
Volume = \( 3.14 \times 7^2 \times 10 = 3.14 \times 49 \times 10 = 1538.6 \ \text{cm}^3 \)
Example:
A cylindrical water tank has a height of 2.5 m and diameter 1.4 m. Calculate the total surface area and volume in cubic metres. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Radius = \( \frac{1.4}{2} = 0.7 \ \text{m} \)
CSA = \( 2 \times 3.14 \times 0.7 \times 2.5 = 10.99 \ \text{m}^2 \)
Area of bases = \( 2 \times 3.14 \times 0.7^2 = 2 \times 3.14 \times 0.49 = 3.0788 \ \text{m}^2 \)
Total Surface Area = \( 10.99 + 3.0788 = 14.07 \ \text{m}^2 \)
Volume = \( 3.14 \times 0.7^2 \times 2.5 = 3.14 \times 0.49 \times 2.5 = 3.84725 \ \text{m}^3 \)
Sphere
Sphere
A sphere is a perfectly round 3D shape where every point on the surface is equidistant from the center. It has no edges or vertices.
Surface Area
\( \text{Surface Area} = 4 \pi r^2 \)
Volume
\( \text{Volume} = \frac{4}{3} \pi r^3 \)
Example:
A sphere has radius 5 cm. Calculate its surface area and volume. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Surface Area = \( 4 \times 3.14 \times 5^2 = 4 \times 3.14 \times 25 = 314 \ \text{cm}^2 \)
Volume = \( \frac{4}{3} \times 3.14 \times 5^3 = \frac{4}{3} \times 3.14 \times 125 = \frac{1570}{3} \approx 523.33 \ \text{cm}^3 \)
Example:
The surface area of a sphere is 804.25 m². Find the radius of the sphere. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Using \( 4 \pi r^2 = 804.25 \)
\( r^2 = \frac{804.25}{4 \times 3.14} = \frac{804.25}{12.56} \approx 64 \)
\( r = \sqrt{64} = 8 \ \text{m} \)
Pyramid
Pyramid
A pyramid is a 3D shape with a polygonal base and triangular faces that meet at a common vertex (the apex). The most common is the square-based pyramid.
Surface Area
- Lateral Surface Area: $\text{Sum of the areas of the triangular faces}$
- Total Surface Area: $\text{Base area + lateral surface area}$
Volume
\( \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \)
Example:
A square-based pyramid has base side 6 cm and vertical height 9 cm. Find its volume.
▶️ Answer/Explanation
Base Area = \( 6 \times 6 = 36 \ \text{cm}^2 \)
Volume = \( \frac{1}{3} \times 36 \times 9 = \frac{1}{3} \times 324 = 108 \ \text{cm}^3 \)
Example:
A square pyramid has base side 10 m and slant height 13 m. Calculate its total surface area.
▶️ Answer/Explanation
Base Area = \( 10 \times 10 = 100 \ \text{m}^2 \)
Each triangular face area = \( \frac{1}{2} \times 10 \times 13 = 65 \ \text{m}^2 \)
Total of 4 triangles = \( 4 \times 65 = 260 \ \text{m}^2 \)
Total Surface Area = \( 100 + 260 = 360 \ \text{m}^2 \)
Cone
Cone
A cone is a 3D shape with a circular base and a curved surface that tapers smoothly to a point (the apex). It has one flat face and one curved surface.
Surface Area
- Curved Surface Area: \( \pi r l \), where \( l \) is the slant height
- Total Surface Area: \( \pi r^2 + \pi r l \)
Volume
\(\text{Volume} = \frac{1}{3} \pi r^2 h \)
Example:
Find the total surface area of a cone with radius 7 cm and slant height 25 cm. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Curved Surface Area = \( \pi r l = 3.14 \times 7 \times 25 = 549.5 \ \text{cm}^2 \)
Base Area = \( \pi r^2 = 3.14 \times 7^2 = 3.14 \times 49 = 153.86 \ \text{cm}^2 \)
Total Surface Area = \( 549.5 + 153.86 = 703.36 \ \text{cm}^2 \)
Example:
A cone has radius 6 cm and vertical height 12 cm. Calculate its volume. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Volume = \( \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 6^2 \times 12 \)
= \( \frac{1}{3} \times 3.14 \times 36 \times 12 = \frac{1}{3} \times 1356.48 = 452.16 \ \text{cm}^3 \)