Surface Area and Volume of 3D Solids

A sphere has a radius of $7\text{ cm}$. Calculate its volume and surface area to 1 decimal place.

Step 1: Substitute the radius ($r = 7$) into the volume formula.

• $V = \frac{4}{3} \times \pi \times 7^3$

Step 2: Calculate the volume and state the units.

• $V = 1436.755... \approx 1436.8\text{ cm}^3$

Step 3: Substitute the radius into the surface area formula.

• $SA = 4 \times \pi \times 7^2$

Step 4: Calculate the surface area and state the units.

• $SA = 615.752... \approx 615.8\text{ cm}^2$

The volume of a sphere is $12,000\text{ cm}^3$. Calculate the radius $r$ to 1 decimal place.

Step 1: Set the volume formula equal to the given volume.

• $12000 = \frac{4}{3}\pi r^3$

Step 2: Rearrange to isolate $r^3$.

• $r^3 = \frac{36000}{4\pi} = \frac{9000}{\pi}$

Step 3: Cube root to find $r$.

• $r = \sqrt[3]{\frac{9000}{\pi}} \approx 14.2\text{ cm}$

Pyramids

The Great Pyramid of Giza was built over 4,500 years ago, yet the mathematics used to find its volume remains identical today. A pyramid is a 3D shape with a polygon base and flat triangular faces that meet at a common point called the apex. The vertical distance from the centre of the base to the apex is the perpendicular height ($h$).

You must memorise the volume of a pyramid, as it is often not provided in the exam:

$V = \frac{1}{3} \times \text{Area of Base} \times h$

A square-based pyramid has a base side length of $6\text{ cm}$ and a perpendicular height of $10\text{ cm}$. Calculate its volume.

Step 1: Calculate the area of the square base.

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

Step 2: Substitute the base area and perpendicular height into the volume formula.

• $V = \frac{1}{3} \times 36 \times 10$

Step 3: Calculate the final volume and state the units.

• $V = 12 \times 10 = 120\text{ cm}^3$

To find the surface area, you calculate the area of the base and add the area of the triangular faces. To do this, you often need the slant height ($l$), which is the distance from the edge of the base to the apex along the face of the triangle. You can use Pythagoras' Theorem ($a^2 + b^2 = c^2$) to convert between perpendicular height and slant height.

A square-based pyramid has a base side length of $10\text{ cm}$ and a perpendicular height of $12\text{ cm}$. Calculate its total surface area.

Step 1: Use Pythagoras' Theorem to find the slant height ($l$). The base of the right-angled triangle is half the pyramid's base ($5\text{ cm}$).

• $l^2 = 12^2 + 5^2$
• $l = \sqrt{144 + 25} = \sqrt{169} = 13\text{ cm}$

Step 2: Calculate the area of one triangular face.

• $\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height}$
• $\text{Area}=\frac{1}{2}\times$

Step 3: Calculate the area of all four triangular faces (lateral area).

• $\text{Lateral Area} = 4 \times 65 = 260\text{ cm}^2$

Step 4: Calculate the base area and add it to the lateral area for the total surface area.

• $\text{Base Area} = 10 \times 10 = 100\text{ cm}^2$
• $TSA = 100 + 260 = 360\text{ cm}^2$

Cones

You can think of a cone as a pyramid with a circular base, meaning its volume follows the exact same "one-third" rule. If the top portion of a cone or pyramid is cut off, the remaining shape is called a frustum.

The formulas for a cone are typically provided in the exam:

$V = \frac{1}{3}\pi r^2 h$

$CSA = \pi r l$

The Total Surface Area ($TSA$) is the curved surface area plus the circular base ($TSA = \pi r l + \pi r^2$). Just like with pyramids, the perpendicular height ($h$), radius (), and slant height () form a right-angled triangle, meaning .

A cone has a radius of $5.7\text{ cm}$ and a slant height of $12.6\text{ cm}$. Calculate its volume to 3 significant figures.

Step 1: Use Pythagoras' Theorem to find the perpendicular height ($h$).

• $h = \sqrt{12.6^2 - 5.7^2}$

Step 2: Substitute into the volume formula.

• $V = \frac{1}{3} \times \pi \times 5.7^2 \times 11.237$

Step 3: Calculate the final volume and round to 3 s.f.

• $V = 382.3... \approx 382\text{ cm}^3$

Composite Solids

Most real-world objects aren't simple geometric shapes, but are built by combining multiple basic 3D shapes together to form a composite solid.

To find the volume of a composite solid, simply calculate the volume of each individual shape and add them together (or subtract if a piece has been removed). Finding the surface area is trickier: you must only calculate the exposed faces. Any internal faces where two shapes are joined together are hidden and must not be included in your final sum.

A composite solid is formed by joining a cone to a hemisphere. The radius of both shapes is $5\text{ cm}$ and the perpendicular height of the cone is $12\text{ cm}$. Calculate the total volume and total surface area in terms of $\pi$.

Step 1: Calculate the volume of the cone and the hemisphere separately.

• $V_{\text{cone}} = \frac{1}{3} \times \pi \times 5^2 \times 12 = 100\pi\text{ cm}^3$

Step 2: Add the volumes together.

• $V_{\text{total}} = 100\pi + \frac{250}{3}\pi = \frac{550}{3}\pi\text{ cm}^3$

Step 3: For surface area, first find the slant height ($l$) of the cone.

• $l = \sqrt{12^2 + 5^2} = \sqrt{169} = 13\text{ cm}$

Step 4: Calculate the exposed surface areas (ignore the flat circular base joining them).

• $CSA_{\text{cone}} = \pi \times 5 \times 13 = 65\pi\text{ cm}^2$

Step 5: Add the exposed surface areas together.

• $SA_{\text{total}} = 65\pi + 50\pi = 115\pi\text{ cm}^2$