Surface Area of Composite Solids
A composite solid is made by joining simple solids, such as a cone on a cylinder or a hemisphere on each end of a cylinder. Its surface area is the sum of the outside faces only, so any surfaces that are joined together are left out. This NSW Year 10 Advanced maths subtopic brings together prisms, cylinders, pyramids, cones and spheres (NESA outcome MA5‑ARE‑P‑01).
You will find surface areas of composite prisms, a buoy made of two cones, a capsule-shaped water tank, a toy rocket, an observatory with a dome and a trophy with a cylinder on a cube. You will deal with holes and grooves that add new inside surfaces, and cost the glass on an office tower. Harder questions solve for an unknown radius or height and compare a solid with a cone added and with a cone hollowed out. It is part of Year 10 Advanced Measurement.
Every question with a fully worked solution.
- Surface Area of Composite Solids Watch
Theory
The surface area of a composite solid is the total area of the faces on the outside: add each visible surface and leave out the faces hidden where the parts join. This Year 10 Advanced maths page covers prisms with composite ends, solids built from cylinders, cones and hemispheres, holes and grooves, and algebraic problems.
A composite solid is made by joining two or more simple solids, or by cutting one solid out of another.
Its surface area is the area of everything a coat of paint would touch. When two solids are glued together, the faces pressed against each other are hidden and are not counted.
When a smaller shape sits on a larger face (a cylinder on a cube, one tier of a cake on another), the circle it covers is lost but its own top adds the same area back. So the visible tops add up to the top face of the larger solid.
A hole or groove removes some flat area and adds a new curved inside surface. Always subtract what is cut away and add what is exposed.
The curved surface of a hemisphere (half a sphere) is \(2\pi r^2\). Two equal hemispheres on the ends of a cylinder make one whole sphere, \(4\pi r^2\).
Surface area of a composite solid:
Pieces you will need:
| Part | Surface | Area |
|---|---|---|
| Cylinder | curved wall | \(2\pi rh\) |
| Cone | curved surface | \(\pi rl\), with \(l^2 = r^2 + h^2\) |
| Hemisphere | curved surface (half a sphere) | \(2\pi r^2\) |
| Square pyramid | four triangles | \(4 \times \dfrac{1}{2} \times b \times s\) |
| Any prism | all the side rectangles | perimeter of the end \(\times\) length |
| Half a cylinder | curved part | \(\pi rh\) |
How to find the surface area of a composite solid
- Split the solid into simple parts and sketch each one.
- List every surface on the outside. Cross out faces that are joined, covered or open.
- Find any missing lengths: slant heights with Pythagoras' theorem, radii from diameters.
- Calculate each area and add, keeping \(\pi\) or surds exact until the end.
Special cases
- Shape on a face (cylinder on a cube, pyramid on a cube, stacked tiers): the covered patch and the new top cancel for circles on flat faces; count only five faces of a cube under a pyramid.
- Hole through a solid: subtract the two openings, add the inside wall.
- Groove cut along a face: subtract the strip of the face that is removed and the two end pieces; add the curved surface of the groove.
- Something added versus the same shape hollowed out: the outside of a cone and the inside of the same cone have equal area, \(\pi rl\).
- Algebraic versions: write every surface in terms of the pronumeral, collect like terms, then solve. A quadratic may need factorising.
- Costs: include only the surfaces that are painted, glazed or covered, then multiply by the rate.
A prism is two ends plus a rectangle for every edge of the end. All those rectangles together are (perimeter of the end) \(\times\) length.
| \(\text{one end}\) | \(=\) | \(8 \times 6 - 5 \times 3 = 33\text{ cm}^2\) |
| \(\text{perimeter}\) | \(=\) | \(8 + 3 + 5 + 3 + 3 + 6 = 28\text{ cm}\) |
| \(\text{SA}\) | \(=\) | \(2 \times 33 + 28 \times 10\) |
| \(\) | \(=\) | \(66 + 280\) |
| \(\) | \(=\) | \(\mathbf{346}\text{ cm}^2\) |
The perimeter of an L shape equals the perimeter of the rectangle around it: \(2(8 + 6) = 28\) ✓
The outside is the base, the curved wall and the cone's curved surface. The circle where they join is hidden.
| \(l^2\) | \(=\) | \(9^2 + 12^2 = 225, \ l = 15\) |
| \(\text{SA}\) | \(=\) | \(\pi r^2 + 2\pi rh + \pi rl\) |
| \(\) | \(=\) | \(81\pi + 2\pi(9)(15) + \pi(9)(15)\) |
| \(\) | \(=\) | \(81\pi + 270\pi + 135\pi\) |
| \(\) | \(=\) | \(\mathbf{486\pi}\text{ cm}^2\) |
| \(\) | \(=\) | \(1526.8140\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{1526.81}\text{ cm}^2\) |
List the surfaces: the outer curved wall, the inner curved wall, and two flat rings (a big circle minus a small circle at each end).
| \(\text{outer wall}\) | \(=\) | \(2\pi(5)(12) = 120\pi\) |
| \(\text{inner wall}\) | \(=\) | \(2\pi(2)(12) = 48\pi\) |
| \(\text{two rings}\) | \(=\) | \(2(\pi \times 5^2 - \pi \times 2^2) = 42\pi\) |
| \(\text{SA}\) | \(=\) | \(120\pi + 48\pi + 42\pi\) |
| \(\) | \(=\) | \(\mathbf{210\pi}\text{ cm}^2\) |
| \(\) | \(\approx\) | \(\mathbf{659.73}\text{ cm}^2\) |
Drilling a hole can increase the surface area: the inside wall (\(48\pi\)) is bigger than the two circles removed (\(8\pi\)).
(i) Show that its surface area is \(4\pi r^2\).
(ii) The surface area is \(100\pi\text{ cm}^2\). Find \(r\) and the exact perpendicular height of the cone.
(i) The flat faces are joined, so only the two curved surfaces count. The curved surface of a hemisphere is half of a sphere's \(4\pi r^2\):
| \(\text{SA}\) | \(=\) | \(2\pi r^2 + \pi r(2r)\) |
| \(\) | \(=\) | \(2\pi r^2 + 2\pi r^2 = 4\pi r^2\) |
(ii)
| \(4\pi r^2\) | \(=\) | \(100\pi\) |
| \(r^2\) | \(=\) | \(25\) |
| \(r\) | \(=\) | \(\mathbf{5} \quad (r \gt 0)\) |
| \(h^2\) | \(=\) | \(10^2 - 5^2 = 75\) |
| \(h\) | \(=\) | \(\sqrt{25 \times 3} = \mathbf{5\sqrt{3}}\text{ cm}\) |
Common pitfalls
Frequently asked questions
How do you find the surface area of a composite solid?
Split it into simple solids, list every surface that is on the outside, leave out faces that are joined or hidden, find each area and add them.
Do you include the area where two solids are joined?
No. The two faces pressed together are inside the solid, so neither is part of the surface area.
What is the surface area of a hemisphere?
The curved surface of a hemisphere is \(2\pi r^2\), half of a sphere's \(4\pi r^2\). A solid hemisphere also has a flat circle, \(\pi r^2\), so its total is \(3\pi r^2\).
Does drilling a hole increase or decrease the surface area?
It can do either. You lose the two openings but gain the inside wall of the hole. For a long, narrow hole the inside wall is usually larger, so the surface area goes up.
How do you find the surface area of a prism with an L-shaped or T-shaped end?
Find the area of one end and its perimeter. The surface area is twice the end area plus the perimeter times the length of the prism.