IB MYP 4-5 Maths- Volume and capacity- Study Notes - New Syllabus
IB MYP 4-5 Maths- Volume and capacity – Study Notes
Extended
- Volume and capacity
IB MYP 4-5 Maths- Volume and capacity – Study Notes – All topics
Volume and Capacity of Known Shapes
Volume and Capacity of Known Shapes
Volume is the amount of 3D space occupied by a solid. It is measured in cubic units (e.g., cm³, m³). Capacity refers to the amount of space available inside a container, usually expressed in liters (L) or milliliters (mL).
Key Conversion:
- \( 1 \,\text{cm}^3 = 1 \,\text{mL} \)
- \( 1000 \,\text{cm}^3 = 1 \,\text{L} \)
- \( 1 \,\text{m}^3 = 1000 \,\text{L} \)
Capacity
- 1 liter = 1000 milliliters = 1000 cm³
- 1 m³ = 1000 liters
- To convert volume to capacity, use these relationships.
Cube
A cube is a regular 3D solid with 6 equal square faces, 12 equal edges, and 8 vertices. All edges have the same length \( a /s\).
Formulas:
- Volume: \( V = a^3 \)
- Total Surface Area (TSA): \( \text{TSA} = 6a^2 \)
- Lateral Surface Area (LSA): \( \text{LSA} = 4a^2 \)
Example:
Find the volume and surface area of a cube of edge 5 cm.
▶️ Answer/Explanation
Step 1: Volume
\( V = a^3 = 5^3 = 125 \,\text{cm}^3 \).
Step 2: TSA
\( \text{TSA} = 6a^2 = 6(5^2) = 6 \times 25 = 150 \,\text{cm}^2 \).
Example:
A cube-shaped container has an edge of 20 cm. Find its capacity in liters and its lateral surface area.
▶️ Answer/Explanation
Step 1: Volume
\( V = a^3 = 20^3 = 8000 \,\text{cm}^3 \).
Convert to liters: \( 8000 \div 1000 = 8 \,\text{L} \).
Step 2: LSA
\( \text{LSA} = 4a^2 = 4(20^2) = 4 \times 400 = 1600 \,\text{cm}^2 \).
Cuboid
A cuboid is a 3D solid with 6 rectangular faces. Dimensions: length (\( l \)), breadth (\( b \)), and height (\( h \)).
Formulas:
- Volume: \( V = l \times b \times h \)
- Total Surface Area (TSA): \( \text{TSA} = 2(lb + bh + hl) \)
- Lateral Surface Area (LSA): \( \text{LSA} = 2h(l + b) \)
Example:
Find the volume and total surface area of a cuboid of dimensions 10 cm × 6 cm × 4 cm.
▶️ Answer/Explanation
Step 1: Volume
\( V = l \times b \times h = 10 \times 6 \times 4 = 240 \,\text{cm}^3 \).
Step 2: TSA
\( \text{TSA} = 2(lb + bh + hl) = 2(10 \times 6 + 6 \times 4 + 4 \times 10) = 2(60 + 24 + 40) = 248 \,\text{cm}^2 \).
Example:
A fish tank is in the shape of a cuboid measuring 50 cm × 30 cm × 40 cm. Find its capacity in liters and lateral surface area.
▶️ Answer/Explanation
Step 1: Volume
\( V = 50 \times 30 \times 40 = 60,000 \,\text{cm}^3 \).
Convert to liters: \( 60,000 \div 1000 = 60 \,\text{L} \).
Step 2: LSA
\( \text{LSA} = 2h(l + b) = 2 \times 40(50 + 30) = 2 \times 40 \times 80 = 6,400 \,\text{cm}^2 \).
Cylinder
A cylinder is a 3D solid with two parallel circular bases connected by a curved surface. Defined by radius \( r \) and height \( h \).
Formulas:
- Volume: \( V = \pi r^2 h \)
- Total Surface Area (TSA): \( \text{TSA} = 2\pi r(h + r) \)
- Curved Surface Area (CSA): \( \text{CSA} = 2\pi r h \)
Example:
Find the volume, curved surface area, and total surface area of a cylinder with radius 7 cm and height 10 cm. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Step 1: Volume
\( V = \pi r^2 h = 3.14 \times (7)^2 \times 10 = 3.14 \times 49 \times 10 = 1538.6 \,\text{cm}^3 \).
Step 2: Curved Surface Area
\( \text{CSA} = 2\pi r h = 2 \times 3.14 \times 7 \times 10 = 439.6 \,\text{cm}^2 \).
Step 3: Total Surface Area
\( \text{TSA} = 2\pi r(h + r) = 2 \times 3.14 \times 7(10 + 7) = 2 \times 3.14 \times 7 \times 17 = 747.46 \,\text{cm}^2 \).
Example:
A cylindrical water tank has radius 35 cm and height 100 cm. Find its volume in liters and its total surface area. Take \( \pi = 3.14 \).
▶️ Answer/Explanation
Step 1: Volume
\( V = \pi r^2 h = 3.14 \times (35)^2 \times 100 = 3.14 \times 1225 \times 100 = 384,650 \,\text{cm}^3 \).
Convert to liters: \( 384,650 \div 1000 = 384.65 \,\text{L} \).
Step 2: TSA
\( \text{TSA} = 2\pi r(h + r) = 2 \times 3.14 \times 35(100 + 35) = 2 \times 3.14 \times 35 \times 135 = 29,694 \,\text{cm}^2 \).
Cone
A cone is a 3D solid with a circular base and a curved surface that tapers to a single vertex.
Important dimensions:
- \( r \) = radius of base
- \( h \) = vertical height
- \( l \) = slant height
Formulas:
- Slant Height: \( l = \sqrt{r^2 + h^2} \)
- Volume: \( V = \dfrac{1}{3}\pi r^2 h \)
- Total Surface Area (TSA): \( \text{TSA} = \pi r^2 + \pi r l \)
- Curved Surface Area (CSA): \( \text{CSA} = \pi r l \)
Example:
Find the volume and total surface area of a cone with radius 7 cm and height 24 cm. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Step 1: Find slant height
\( l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \,\text{cm}\).
Step 2: Volume
\( V = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3} \times 3.14 \times (7)^2 \times 24 \).
\( V = \dfrac{1}{3} \times 3.14 \times 49 \times 24 = 1234.6 \,\text{cm}^3 \).
Step 3: TSA
\( \text{TSA} = \pi r^2 + \pi r l = 3.14(7^2) + 3.14(7)(25) = 3.14(49 + 175) = 3.14 \times 224 = 703.36 \,\text{cm}^2 \).
Example:
A conical water tank has a radius of 1.2 m and height 3.6 m. Find:
- Its capacity in liters
- Its curved surface area (CSA)
Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Step 1: Volume (Capacity)
\( V = \dfrac{1}{3}\pi r^2 h \)
\( V = \dfrac{1}{3} \times 3.14 \times (1.2)^2 \times 3.6 \)
\( V = \dfrac{1}{3} \times 3.14 \times 1.44 \times 3.6 \)
\( V = \dfrac{1}{3} \times 16.29 \approx 5.43 \,\text{m}^3 \)
Convert to liters: \( 5.43 \times 1000 = 5430 \,\text{L} \).
Step 2: Slant Height
\( l = \sqrt{r^2 + h^2} = \sqrt{(1.2)^2 + (3.6)^2} = \sqrt{1.44 + 12.96} = \sqrt{14.4} \approx 3.79 \,\text{m} \).
Step 3: Curved Surface Area
\( \text{CSA} = \pi r l = 3.14 \times 1.2 \times 3.79 \approx 14.29 \,\text{m}^2 \).
Sphere and Hemisphere
A sphere is a perfectly round 3D object where every point on its surface is equidistant from the center. A hemisphere is half of a sphere.
Sphere:
- Surface Area (SA): \( \text{SA} = 4\pi r^2 \)
- Volume: \( V = \dfrac{4}{3}\pi r^3 \)
Hemisphere:
- Curved Surface Area (CSA): \( \text{CSA} = 2\pi r^2 \)
- Total Surface Area (TSA): \( \text{TSA} = 3\pi r^2 \)
- Volume: \( V = \dfrac{2}{3}\pi r^3 \)
Example:
A solid sphere has a radius of 7 cm. Find:
- Surface area
- Volume
Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Step 1: Surface Area
\( \text{SA} = 4\pi r^2 = 4 \times 3.14 \times (7)^2 = 4 \times 3.14 \times 49 = 615.44 \,\text{cm}^2 \).
Step 2: Volume
\( V = \dfrac{4}{3}\pi r^3 = \dfrac{4}{3} \times 3.14 \times (7)^3 = \dfrac{4}{3} \times 3.14 \times 343 \approx 1436.76 \,\text{cm}^3 \).
Example:
A hemisphere has radius 10 cm. Find:
- Curved surface area
- Total surface area
- Volume
Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Step 1: CSA
\( \text{CSA} = 2\pi r^2 = 2 \times 3.14 \times (10)^2 = 628 \,\text{cm}^2 \).
Step 2: TSA
\( \text{TSA} = 3\pi r^2 = 3 \times 3.14 \times 100 = 942 \,\text{cm}^2 \).
Step 3: Volume
\( V = \dfrac{2}{3}\pi r^3 = \dfrac{2}{3} \times 3.14 \times (10)^3 = \dfrac{2}{3} \times 3.14 \times 1000 = 2093.33 \,\text{cm}^3 \).
Prism
A prism is a 3D solid with two parallel, congruent polygonal bases and rectangular faces joining them. The shape of the base determines the type of prism (e.g., triangular prism, rectangular prism).
Formulas:
- Lateral Surface Area (LSA): Sum of areas of the rectangular faces excluding the bases.
- Total Surface Area (TSA): \( \text{TSA} = \text{LSA} + 2 \times \text{Base Area} \)
- Volume: \( V = \text{Base Area} \times \text{Height} \)
Example:
A triangular prism has a base triangle with base 8 cm and height 6 cm. The prism length is 10 cm. Find:
- Volume
- Total Surface Area
Use \( \pi = 3.14 \) if needed.
▶️ Answer/Explanation
Step 1: Base Area
\( \text{Base Area} = \dfrac{1}{2} \times 8 \times 6 = 24 \,\text{cm}^2 \).
Step 2: Volume
\( V = \text{Base Area} \times \text{Height (length of prism)} = 24 \times 10 = 240 \,\text{cm}^3 \).
Step 3: TSA
The prism has 2 triangular bases + 3 rectangles. Base area = 24 cm² × 2 = 48 cm². Rectangles: (8 × 10) + (6 × 10) + (10 × 10) = 80 + 60 + 100 = 240 cm².
TSA = 48 + 240 = 288 cm².
Example:
A rectangular prism (cuboid) has length 12 cm, width 8 cm, and height 5 cm. Find:
- Volume
- Total Surface Area
▶️ Answer/Explanation
Step 1: Volume
\( V = l \times w \times h = 12 \times 8 \times 5 = 480 \,\text{cm}^3 \).
Step 2: TSA
\( \text{TSA} = 2(lw + lh + wh) = 2(12 \times 8 + 12 \times 5 + 8 \times 5) \)
\( = 2(96 + 60 + 40) = 2(196) = 392 \,\text{cm}^2 \).
Pyramid
A pyramid is a 3D solid with a polygonal base and triangular faces that meet at a single vertex (apex). The height is the perpendicular distance from the apex to the base.
Formulas:
- Lateral Surface Area (LSA): Sum of the areas of triangular faces.
- Total Surface Area (TSA): \( \text{TSA} = \text{Base Area} + \text{LSA} \)
- Volume: \( V = \dfrac{1}{3} \times \text{Base Area} \times \text{Height} \)
Example :
A square-based pyramid has base edge 10 cm and vertical height 12 cm. Find:
- Volume
- Total Surface Area (TSA), given slant height is 13 cm
▶️ Answer/Explanation
Step 1: Base Area
\( \text{Base Area} = 10 \times 10 = 100 \,\text{cm}^2 \).
Step 2: Volume
\( V = \dfrac{1}{3} \times \text{Base Area} \times \text{Height} = \dfrac{1}{3} \times 100 \times 12 = 400 \,\text{cm}^3 \).
Step 3: Lateral Surface Area
Each triangular face area = \( \dfrac{1}{2} \times \text{base} \times \text{slant height} = \dfrac{1}{2} \times 10 \times 13 = 65 \,\text{cm}^2 \).
4 faces → \( 4 \times 65 = 260 \,\text{cm}^2 \).
Step 4: TSA
TSA = Base Area + LSA = \( 100 + 260 = 360 \,\text{cm}^2 \).
Final Answer: Volume = 400 cm³, TSA = 360 cm².
Example :
A triangular pyramid has an equilateral base of side 6 cm and height 10 cm (from apex to base). Find:
- Volume
▶️ Answer/Explanation
Step 1: Base Area
\( \text{Base Area} = \dfrac{\sqrt{3}}{4} a^2 = \dfrac{\sqrt{3}}{4} (6)^2 = \dfrac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \,\text{cm}^2 \approx 15.59 \,\text{cm}^2 \).
Step 2: Volume
\( V = \dfrac{1}{3} \times \text{Base Area} \times \text{Height} = \dfrac{1}{3} \times 15.59 \times 10 \approx 51.97 \,\text{cm}^3 \).