# Volume and Surface Area | Maths with Lemon

Source: https://mathswithlemon.io/aa-geometry-volume-surface-area.html


### Volume and Surface Area

IB Mathematics: Analysis and Approaches · SL 3.1 (SL/HL) · Lesson 6.2. Learn to calculate the volume and surface area of pyramids, cones, spheres, hemispheres and composite solids, using right-angled triangles to find missing dimensions.

Lesson 6.2 worksheet (PDF) AA Geometry & Trigonometry Unit Circle


#### 1. Volume and Surface Area: The Big Picture


##### What you need to know

- Volume measures the space inside a solid (cubic units).
- Surface area measures its exposed outer area (square units).
- Distinguish perpendicular height from slant height.
- Identify hidden right-angled triangles in 3D shapes.

##### Key Points

For a composite solid, add the volumes of its constituent parts, or subtract a removed volume. For surface area, count only the exposed faces ; joined faces inside the object are not included.


##### Cone

Radius, vertical height, slant height


##### Sphere

All points are a radius from the centre


##### Pyramid

Base area and perpendicular height


#### 2. Right Pyramids


##### What you need to know

- Volume is one-third of the corresponding prism.
- For a right pyramid, the apex is directly above the centre of the base.
- Use Pythagoras to calculate a triangular face’s slant height.
- Add the base area and all triangular faces for total surface area.
Worksheet: Examples 1–2, pages 1–2.


##### Key Formula

Square base side = 12 cm; perpendicular height = 8 cm.

Step 1. Base area: \(12^2=144\text{ cm}^2\).

Step 2. Face slant height, using half the base side:

Step 3. Four triangular faces, each with area \(\frac12(12)(10)\):

Notice that the numerical values of area and volume happen to be equal here, but their units are different.


#### 3. Right Cones and Frustums


##### What you need to know

- Use the perpendicular height for volume.
- Use the slant height for curved surface area.
- A frustum is a large cone with a smaller similar cone removed.
- Check whether a question includes the circular base.
Worksheet: Examples 3–6 and 12.


##### Key Formulas

A cone has radius 3 cm and perpendicular height 7 cm. Find its exact volume.

Keep \(\pi\) in the answer when the question asks for an exact value.

Outer radius = 15 mm; top radius = 3 mm. The large and removed cones have heights 36 mm and 7.2 mm.

The nozzle has no base, so subtract the curved areas only:


#### 4. Spheres and Hemispheres


##### What you need to know

- A sphere has no flat faces.
- A hemisphere has half the sphere’s volume.
- The total surface area of a hemisphere includes its circular base.
- Check whether a question asks for curved or total surface area.
Worksheet: Examples 7–8.


##### Formula Collection

A hemisphere has diameter 20 cm, so its radius is 10 cm.


#### 5. Combined Solids


##### What you need to know

- Split a solid into familiar shapes.
- Add volumes of attached solids.
- Subtract volumes of holes or cut-out sections.
- For surface area, exclude shared internal faces and include newly exposed cut surfaces.
Worksheet: ice-cream model and Examples 9–11, page 4–5.


##### Key Strategy

Draw a quick labelled sketch and mark which surfaces are visible. The formula for volume and the formula for exposed area are often not built from the same set of pieces.

A cone of height 8 cm supports a hemisphere of radius 4 cm.

Volume: add cone and hemisphere.

Surface area: only the cone’s curved surface and the hemisphere’s curved surface are exposed.


#### 6. Formula Explorer


##### Try different dimensions

- Choose a solid.
- Adjust the radius or base side and perpendicular height.
- Observe how the volume and total surface area change.
- For a cone or pyramid, the slant height is calculated automatically.
The calculator assumes a closed solid, including its base.


##### Interactive calculation


#### 7. Practice and IB-Style Challenges


##### From your worksheet

- Examples 1–2: square and hexagonal pyramids.
- Examples 3–6: cones, slant height and a piping nozzle.
- Examples 7–8: sphere and hemisphere.
- Examples 9–12: combined solids and a frustum.
- Challenges 1–2: capsule redesign and fencepost manufacturing.

##### Revision checklist

Can you distinguish height and slant height, use Pythagoras in a 3D context, identify exposed surfaces, and explain your assumptions in an applied problem?

Slant height \(\ell=13\) cm.

Open the complete Lesson 6.2 worksheet


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