Volume And Surface Area
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Volume of Cubes and Cuboids
- Volume is the amount of space a 3D shape takes up.
- For a cuboid with length l, width w, height h: V = lwh.
- A cube is a special cuboid with all sides equal: V = s3.
- Units are cubic, e.g. cm³, m³.
Cuboid
Volume of Prisms
- A prism has a constant cross-sectional area A and length l.
- Volume formula: V = A l (given in exam).
- The cross-section can be any shape (e.g. triangle, L-shape).
- Find the area of the cross-section first, then multiply by length.
Prism
Volume of Cylinders
- A cylinder is like a prism with a circular base.
- Volume: V = \π r2 h (given in exam).
- r = radius, h = height.
Cylinder
Volume of Pyramids and Cones
- Pyramid volume: V = \frac{1}{3} A h (given), where A = base area, h = perpendicular height.
- Cone volume: V = \frac{1}{3} \π r2 h (given).
- The height must be perpendicular to the base.
Pyramid
Volume of Spheres
- Sphere volume: V = \frac{4}{3} \π r3 (given).
- A hemisphere is half a sphere: V = \frac{2}{3} \π r3.
Sphere
Problem Solving with Volumes
- Real-life problems often combine volume with other topics (e.g. money).
- For compound shapes, split into standard solids and add volumes.
- For fractions of shapes (e.g. hemisphere), find the full volume then take the fraction.
- Always check units and round as required (e.g. 3 significant figures).
Compound L-shaped prism
Surface Area of Cuboids and Prisms
- Surface area is the sum of areas of all faces.
- For a cuboid: SA = 2(lw + lh + wh).
- Drawing a net helps identify all faces.
- For prisms, calculate area of each face (including bases) and add.
Cuboid
Surface Area of Cylinders
- Curved surface area: A = 2\π r h (given).
- Total surface area: SA = 2\π r h + 2\π r2 (not given).
- The net is two circles and a rectangle.
Net of a cylinder
Surface Area of Cones and Spheres
- Cone curved surface: A = \π r l (given), l = slant height.
- Cone total: SA = \π r l + \π r2.
- Sphere surface: A = 4\π r2 (given).
- Hemisphere: SA = 2\π r2 + \π r2 = 3\π r2.
Surface area of a hemisphere
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Câu hỏi luyện tập
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1.What is the formula for the volume of a cuboid with length l, width w, and height h?
Easy- AV = lwh
- BV = 2lw + 2lh + 2wh
- CV = l + w + h
- DV = l2 + w2 + h2
2.A cube has side length 8 cm. What is its surface area?
Easy- A384 cm2
- B64 cm2
- C512 cm2
- D192 cm2
3.A cuboid measures 5 cm by 7 cm by 9.5 cm. What is its surface area?
Medium- A298 cm2
- B346 cm2
- C270.75 cm2
- D332.5 cm2
4.A cylinder has radius 6 cm and height 17 cm. What is its volume? (Use π ≈ 3.142 or calculator)
Medium- A1923 cm3
- B1922 cm3
- C1924 cm3
- D1920 cm3
5.A cone has radius 4.5 cm and height 10.4 cm. What is its volume in terms of π?
Medium- A70.2π cm3
- B70.2π cm2
- C210.6π cm3
- D23.4π cm3
6.A cube has a volume of 1000 cm3. What is its surface area?
Medium- A600 cm2
- B100 cm2
- C500 cm2
- D400 cm2
7.A sphere has radius 5.2 cm. What is its surface area? (Use π ≈ 3.142)
Medium- A339.8 cm2
- B339.8 cm3
- C340 cm2
- D339 cm2
8.The volume of a cuboid is 180 cm3. Its base is a square of side 6 cm. What is its height?
Medium- A5 cm
- B30 cm
- C6 cm
- D3 cm
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