Surface Area of Pyramids, Cones, and Frustums

By the end of this lesson, learners should be able to

1. Calculate the total surface area of right pyramids using base area and triangular face areas.
2. Determine the surface area of cones using base radius, perpendicular height, and slant height.
3. Calculate the total surface area of pyramid frustums by summing base areas and trapezoidal side face areas.

Key vocabulary

Pyramid

A three-dimensional solid with a polygonal base and triangular sides meeting at an apex.

The surface area of a square pyramid includes one square base and four triangles.

Slant height

The perpendicular distance along a slanted face from the base to the apex.

A pyramid with a slant height of 20 cm has large triangular side faces.

Cone

A solid figure with a circular base and a single curved surface reaching an apex.

A cone with base radius 21 cm has a circular face and curved surface.

Curved surface area

The area of the non-flat side face of a curved three-dimensional geometric figure.

The curved surface area of this cone with radius 21 cm is 2310 cm².

Frustum

The bottom part of a pyramid or cone left after cutting the top off.

The surface area of the frustum includes both bases and four side faces.

Trapezoid

A quadrilateral having one pair of parallel sides of different lengths.

Each side face is a trapezoid with parallel lengths of 10 cm and 18 cm.

Net

A two-dimensional flat pattern that folds to form a three-dimensional solid.

Unfolding the open frustum net shows four congruent trapezia and one square base.

Lesson preview

Slide 1 · Surface Area of Right PyramidsOpening · 10 min

To find the surface area of a right pyramid, calculate the area of its base and add the total area of its sloping triangular faces. Unfolding the pyramid into its flat net clearly displays each individual face. For a regular square-based pyramid with side length $s$ and slant height $l$, the four side triangles are identical, each having area $\frac{1}{2}sl$. Summing the square base area $s^2$ and the four triangular face areas gives the total surface area.

• A right pyramid has a polygonal base and triangular faces meeting at a top vertex.
• The total surface area of a pyramid equals base area plus area of triangular faces.
• Slant height is the altitude of a triangular face, measured perpendicular to base edge.
• Unfolding a square pyramid net reveals one square base and four congruent triangular faces.

Worked example. Step 1: Calculate the area of the square base: $\text{Base area}=14\text{ cm}\times 14\text{ cm}=196{\text{ cm}}^2$. Step 2: Calculate the area of one triangular face: $\text{Triangle area}=\frac{1}{2}\times 14\text{ cm}\times 20\text{ cm}=140{\text{ cm}}^2$. Step 3: Calculate the area of all four triangular faces: $4\times 140{\text{ cm}}^2=560{\text{ cm}}^2$. Step 4: Add base area and total triangular area: $\text{Total Surface Area}=196{\text{ cm}}^2+560{\text{ cm}}^2=756{\text{ cm}}^2$. Answer: $756{\text{ cm}}^2$.

Check question: A right pyramid has a square base of side $8\text{ cm}$ and a slant face height of $12\text{ cm}$. What is its total surface area?

1. $112{\text{ cm}}^2$
2. $192{\text{ cm}}^2$
3. $256{\text{ cm}}^2$
4. $448{\text{ cm}}^2$

Show answer

$256{\text{ cm}}^2$. The base area is $8\times 8=64{\text{ cm}}^2$. The area of the four triangular side faces is $4\times (\frac{1}{2}\times 8\times 12)=192{\text{ cm}}^2$. The total surface area is $64+192=256{\text{ cm}}^2$.

Slide 2 · Surface Area of ConesModel · 10 min

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This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• All 4 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 8 lessons in this sub-strand

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All 9 lessons in Surface Area and Volume of Solids

1. 1Surface Area of Prisms and PyramidsDetermine the surface area of prisms, pyramids, cones, frustums and spheres
2. 2Surface Area of Pyramids, Cones, and FrustumsDetermine the surface area of prisms, pyramids, cones, frustums and spheres
3. 3Surface Area of Frustums, Spheres, and HemispheresDetermine the surface area of prisms, pyramids, cones, frustums and spheres
4. 4Surface Area of Composite SolidsCalculate the surface area of composite solids
5. 5Volume of Prisms, Pyramids, and ConesCalculate the volume of prisms, pyramids, cones, frustums and spheres
6. 6Volume of Frustums and SpheresCalculate the volume of prisms, pyramids, cones, frustums and spheres
7. 7Volume of Composite SolidsDetermine the volume of composite solids
8. 8Surface Area and Volume of Solids: Real-Life Applications (Part 1)Explore the use of the surface area and volume of solids in real-life situations
9. 9Real-Life Applications of Surface Area and Volume of SolidsExplore the use of the surface area and volume of solids in real-life situations