Right square-pyramid surface area
Use the square base and four congruent triangular faces. The height of a triangular face is its slant height
\[SA=b^2+2bs\]
- If the slant height is not given Use the vertical height and half the base side \(\displaystyle s^2=h^2+(b/2)^2\)
- If the base is absent Use only the four triangular faces \(\displaystyle A_{\text{faces}}=2bs\)
- Use square units Convert all lengths to one unit before multiplying
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
\[\begin{gathered}b=6\text{ cm},\quad h=4\text{ cm}\\s=5\text{ cm}\end{gathered}\]
- \(\displaystyle A_{\text{base}}=6^2=36\text{ cm}^2\)Calculate the area of the square base
- \(\displaystyle \begin{gathered}A_{\triangle}=\frac12\times6\times5\\{}=15\text{ cm}^2\end{gathered}\)Use the slant height as the perpendicular height of each triangular face
- \(\displaystyle \begin{gathered}A_{\text{faces}}=4\times15\\{}=60\text{ cm}^2\end{gathered}\)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
✕\(\displaystyle \begin{gathered}SA=8^2+4\left(\frac12\times 8\times 3\right)\\{}=112\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=8^2+4\left(\frac12\times 8\times 5\right)\\{}=144\text{ cm}^2\end{gathered}\)
The triangular faces are measured along the face with slant height 5 cm, not vertical height 3 cmLeaving out the square base
✕\(\displaystyle \begin{gathered}SA=4\left(\frac12\times6\times5\right)\\{}=60\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=6^2+4\left(\frac12\times6\times5\right)\\{}=96\text{ cm}^2\end{gathered}\)
Surface area includes the square base as well as all four triangular facesConfusing surface area and volume
✕\(\displaystyle \begin{gathered}V=\frac13\times6^2\times4\\{}=48\text{ cm}^3\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=6^2+4\left(\frac12\times6\times5\right)\\{}=96\text{ cm}^2\end{gathered}\)
Volume uses base area and vertical height, while surface area adds the areas of the boundary facesDifficulty levels
standardProduce the total surface area of a square-based pyramid with its slant height given
examRecognise the surface-area result among same-unit values
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
Is this required Year 9 content?+
This is an optional extension. VCAA places right-pyramid surface-area problems in Level 10A
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 5: Measurement
In this chapter
5A Length conversion and perimeter: Convert measurement units5A Length conversion and perimeter: Perimeter and area of standard shapesFind an unknown side from a perimeter5A Length conversion and perimeter: Solve a linear equation5A Length conversion and perimeter: Substitute values into a formula5A Length conversion and perimeter: Round measurements to decimal places5B Circumference, sectors and exact π: Circumference and area of a circle5B Circumference, sectors and exact π: Arc length and sector areaPerimeter of a sector5B Circumference, sectors and exact π: Simplify exact expressions involving pi5B Circumference, sectors and exact π: Substitute values into a formula5B Circumference, sectors and exact π: Round measurements to decimal places5C Area formulas and units: Convert measurement units5C Area formulas and units: Perimeter and area of standard shapes5C Area formulas and units: Circumference and area of a circle5C Area formulas and units: Arc length and sector area5C Area formulas and units: Simplify exact expressions involving pi5C Area formulas and units: Substitute values into a formula5C Area formulas and units: Round measurements to decimal placesPerimeter and area of composite shapes5D Composite perimeter and area: Find a side of a right-angled triangle5D Composite perimeter and area: Substitute values into a formula5D Composite perimeter and area: Round measurements to decimal places5E Right-prism surface area: Surface area of cuboidsSurface area of a triangular prism5E Right-prism surface area: Find a side of a right-angled triangle5E Right-prism surface area: Substitute values into a formula5E Right-prism surface area: Round measurements to decimal placesPyramid surface area: Surface area of square pyramids (optional extension)5F Cylinder surface area: Surface area of cylindersSurface area of a cylinder portion5F Cylinder surface area: Substitute values into a formula5F Cylinder surface area: Round measurements to decimal places5G Volume and capacity: Convert measurement units5G Volume and capacity: Volume of right prisms5G Volume and capacity: Capacity of a container5G Volume and capacity: Solve a linear equation5G Volume and capacity: Substitute values into a formula5G Volume and capacity: Round measurements to decimal places5H Cylinder volume and capacity: Volume of cylinders5H Cylinder volume and capacity: Capacity of cylindersVolume of a cylinder portion5H Cylinder volume and capacity: Substitute values into a formula5H Cylinder volume and capacity: Round measurements to decimal placesSmallest unit and limits of accuracy