Surface area of solids practice questions | General Mathematics (Year 11, VCE Units 1 & 2)

Surface area of solids

Surface area is found by listing each included face group, calculating its area with the correct dimensions, and adding the results. A slant height is different from a perpendicular vertical height

SA_{\text{pyramid}}=b^2+2bs,\quad SA_{\text{cylinder}}=2\pi r^2+2\pi rh

List the included faces Use the solid's net as a checklist and count every included face once \(\displaystyle SA=b^2+4\left(\frac12bs\right)\)
Use the correct pyramid dimensions Square the base side for the base area; use the base side and slant height for each triangular face \(\displaystyle b=6,\quad s=5\;\Rightarrow\;b^2+2bs=36+60\)
Use the cylinder formula A closed cylinder has two circular ends and one unrolled rectangular curved surface \(\displaystyle SA=2\pi r^2+2\pi rh\)
Keep area units Surface area is measured in square units, and exact π terms are rounded only after they have been added \(\displaystyle SA=36+60=96\text{ cm}^2\)

Worked example

A closed square-based pyramid has base side length 6 cm, vertical height 4 cm and slant height 5 cm. Find its total surface area

b=6\text{ cm},\quad h=4\text{ cm},\quad s=5\text{ cm}

\(\displaystyle A_{\text{base}}=6^2=36\text{ cm}^2\)Calculate the area of the square base
\(\displaystyle A_{\text{one triangle}}=\frac12\times6\times5=15\text{ cm}^2\)Use the slant height as the perpendicular height of each triangular face
\(\displaystyle A_{\text{four triangles}}=4\times15=60\text{ cm}^2\)There are four congruent triangular faces
\(\displaystyle SA=36+60=96\text{ cm}^2\)Add the base and triangular-face areas to obtain the total surface area

Common mistakes

Using vertical height instead of slant height

✕\(\displaystyle SA=8^2+4\left(\frac12\times 8\times 3\right)=112\text{ cm}^2\)✓\(\displaystyle SA=8^2+4\left(\frac12\times 8\times 5\right)=144\text{ cm}^2\)

The triangular faces are measured along the face with slant height 5 cm, not vertical height 3 cm

Leaving out the square base

✕\(\displaystyle SA=4\left(\frac12\times6\times5\right)=60\text{ cm}^2\)✓\(\displaystyle SA=6^2+4\left(\frac12\times6\times5\right)=96\text{ cm}^2\)

Surface area includes the square base as well as all four triangular faces

Confusing surface area and volume

✕\(\displaystyle V=\frac13\times6^2\times4=48\text{ cm}^3\)✓\(\displaystyle SA=6^2+4\left(\frac12\times6\times5\right)=96\text{ cm}^2\)

Volume uses base area and vertical height, while surface area adds the areas of the boundary faces

Difficulty levels

introAdd the face groups of a closed cuboid exactly

standardFind the exact face groups of a closed cylinder or a square-based pyramid with slant height given

challengeHandle open faces, curved solids, a cone with recovered slant height, a hemisphere, or a triangular prism

examChoose the square-pyramid surface area from options containing omitted-face, volume, and slant-height errors

Questions about this topic

What does surface area include?+

It includes the area of every face or curved surface that the question states is included

Why is slant height used on a pyramid face?+

The slant height is perpendicular to the base edge of each triangular face, so it is the height used in that triangle's area formula

How is a closed cylinder's surface area found?+

Add the areas of the two circular ends and the curved surface

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 10: Measurement and similarity

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Surface area of solids