Cones and Spheres

• A hemisphere is exactly half of a sphere, created by slicing it through the centre.
• Its volume is simply half that of a full sphere ($\frac{2}{3}\pi r^3$).
• However, the total surface area of a solid hemisphere requires adding the curved outer surface ($2\pi r^2$) to the new flat circular base ($\pi r^2$), resulting in a total surface area of $3\pi r^2$.

Worked Example: Sphere Calculations

Calculate the volume and surface area of a sphere with a radius of $8.5\,\text{cm}$. Give both answers to 1 decimal place.

Step 1: State the volume formula and substitute the radius.

• $V = \frac{4}{3}\pi r^3$
• $V = \frac{4}{3} \times \pi \times 8.5^3$

Step 2: Calculate the volume.

• $V = \frac{4}{3} \times \pi \times 614.125 \approx 2572.44...$
• Volume = $$

Step 3: State the surface area formula and substitute the radius.

• $A = 4\pi r^2$
• $A = 4 \times \pi \times 8.5^2$

Step 4: Calculate the surface area.

• $A = 4 \times \pi \times 72.25 = 289\pi \approx 907.92...$
• Surface Area = $907.9\,\text{cm}^2$

Worked Example: Hemisphere Surface Area (Exact Value)

A solid hemisphere has a diameter of $18\,\text{mm}$. Find its total surface area in terms of $\pi$.

Step 1: Find the radius.

• $r = 18 \div 2 = 9\,\text{mm}$

Step 2: Use the total surface area formula for a solid hemisphere ($3\pi r^2$).

• $A = 3 \times \pi \times 9^2$

Step 3: Calculate the final exact value.

• $A = 3 \times \pi \times 81$
• Total Surface Area = $243\pi\,\text{mm}^2$

Cones: Volume and Surface Area

• Why do party hats and traffic cones have the same fundamental shape? Their wide bases provide stability, while their tapering points allow for efficient stacking.
• To calculate the properties of a cone, you must carefully distinguish between two different height measurements.
• The perpendicular height ($h$) is the vertical straight-line distance from the tip to the centre of the flat base. This is required for calculating volume.
• The slant height ($l$) is the diagonal distance down the side of the cone to the edge of the circular base. This is required for calculating surface area.
• The radius, perpendicular height, and slant height form a right-angled triangle, meaning you can use Pythagoras' theorem ($r^2 + h^2 = l^2$) to find a missing dimension.

$$\text{Volume of a cone} = \frac{1}{3}\pi r^2 h$$ $$\text{Curved surface area of a cone} = \pi r l$$

• To find the total surface area of a solid cone, you must add the curved surface area to the area of the circular base: $A = \pi r l + \pi r^2$.

Worked Example: Volume of a Cone

A cone has a radius of $3\,\text{cm}$ and a perpendicular height of $10\,\text{cm}$. Calculate its volume to 1 decimal place.

Step 1: State the volume formula.

• $V = \frac{1}{3}\pi r^2 h$

Step 2: Substitute the values for radius ($r = 3$) and perpendicular height ($h = 10$).

• $V = \frac{1}{3} \times \pi \times 3^2 \times 10$

Step 3: Calculate the final volume.

• $V = \frac{1}{3} \times \pi \times 9 \times 10 = 30\pi$

Worked Example: Cone Surface Area (Finding Slant Height)

A solid cone has a radius of $6\,\text{m}$ and a perpendicular height of $8\,\text{m}$. Calculate its total surface area to 3 significant figures.

Step 1: Use Pythagoras' theorem to find the slant height ($l$).

• $l = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100}$

Step 2: Calculate the curved surface area ($\pi r l$).

• $\text{CSA} = \pi \times 6 \times 10 = 60\pi$

Step 3: Calculate the base area ($\pi r^2$).

• $\text{Base} = \pi \times 6^2 = 36\pi$

Step 4: Add them together for the total surface area.

• $\text{Total } A = 60\pi + 36\pi = 96\pi$
• $96\pi \approx 301.59... = 302\,\text{m}^2$

Composite Solids and Frustums

• Designing an ice cream cone with a perfectly spherical scoop on top requires calculating the space of two different joined shapes.
• A composite solid is a 3D figure made of two or more simpler geometric shapes joined together.
• To find the total volume of an additive composite shape, you simply calculate the volume of each individual part and add them together.
• To find the surface area of a composite solid, you must only include the exposed outer faces. Internal faces — where the two shapes touch — are hidden and must be excluded from your calculation.
• A frustum is a truncated solid, formed when the top portion of a cone is sliced off by a plane parallel to the base.
• To find the volume of a frustum or a hollowed-out shape, you calculate the volume of the original large solid and subtract the volume of the removed section. You may need to use a scale factor and similar triangles to find the dimensions of the removed cone.

Worked Example: Composite Volume (Exact Value)

A toy is made by attaching a solid cone of radius $4\,\text{cm}$ and height $9\,\text{cm}$ to a solid hemisphere of radius $4\,\text{cm}$. Find the total volume of the toy, leaving your answer in terms of $\pi$.

Step 1: Calculate the volume of the cone.

• $V_{\text{cone}} = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times \pi \times 4^2 \times 9 = 48\pi\,\text{cm}^3$

Step 2: Calculate the volume of the hemisphere.

• $V_{\text{hemi}} = \frac{2}{3}\pi r^3 = \frac{2}{3} \times \pi \times 4^3 = \frac{128}{3}\pi\,\text{cm}^3$

Step 3: Add the volumes together, using a common denominator.

• $V_{\text{total}} = 48\pi + \frac{128}{3}\pi = \frac{144}{3}\pi + \frac{128}{3}\pi$

Worked Example: Subtractive Volume

A wooden cube has side lengths of $12\,\text{cm}$. A cone with a radius of $5\,\text{cm}$ and a perpendicular height of $8\,\text{cm}$ is completely carved out of the cube. Find the remaining volume of wood to 1 decimal place.

Step 1: Calculate the original volume of the cube.

• $V_{\text{cube}} = 12 \times 12 \times 12 = 1728\,\text{cm}^3$

Step 2: Calculate the volume of the removed cone.

• $V_{\text{cone}} = \frac{1}{3} \times \pi \times 5^2 \times 8 = \frac{200}{3}\pi \approx 209.439...\,\text{cm}^3$

Step 3: Subtract the cone's volume from the cube's volume.

• $V_{\text{remaining}} = 1728 - 209.439... = 1518.56...$
• Remaining Volume = $1518.6\,\text{cm}^3$