Volume of Prisms, Pyramids, and Cones

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

1. Calculate the volume of uniform prisms using cross-sectional area and length
2. Calculate the volume of right pyramids given base dimensions and perpendicular height
3. Calculate the volume of right cones given base radius and perpendicular or slant height

Key vocabulary

Prism

A three-dimensional solid with two identical parallel end faces and uniform cross-section.

The triangular prism has a cross-sectional area of 24 square centimetres.

Cross-section

The shape formed by cutting straight through a solid parallel to its end.

The cross-section of the prism is a right-angled triangle.

Pyramid

A solid with a polygonal base and triangular faces meeting at a vertex.

The rectangular pyramid has a volume equal to one-third of base area times height.

Perpendicular height

The straight line distance from the apex vertically down to the base plane.

The perpendicular height of the pyramid measures fifteen centimetres from base to apex.

Cone

A solid shape with a circular base tapering smoothly to a point.

The cone has a base radius of six centimetres and vertical height of eight.

Slant height

The length measured along the sloping side from base edge to apex.

Using the slant height of ten centimetres gives a perpendicular height of eight.

Lesson preview

Slide 1 · Determining Volume of a PrismOpening · 10 min

The volume of solids measures the total three-dimensional space enclosed within a boundary. For any uniform prism, the cross-section remains identical along its entire length. Determining volume of a prism requires finding the cross-sectional area and multiplying it by the length of the solid. The general formula $V=Ah$ applies to all right prisms, where $A$ represents the area of the uniform cross-section and $h$ is the perpendicular length.

• Volume quantifies the amount of three-dimensional space contained within a closed solid object.
• A prism possesses a uniform cross-section along its entire length from end to end.
• Determining volume of a prism uses the key formula volume equals cross-sectional area times length.
• The formula V equals A h applies to rectangular, triangular, and composite right prisms.

Worked example. Identify the area of the uniform triangular cross-section: $A=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times 8\text{ cm}\times 6\text{ cm}=24{\text{ cm}}^2$. Apply the prism volume formula: $V=A\times h$ where $h$ is the length. Multiply cross-sectional area by length: $V=24{\text{ cm}}^2\times 15\text{ cm}=360{\text{ cm}}^3$. Answer: $360{\text{ cm}}^3$.

Check question: A prism has a uniform trapezoidal cross-section with parallel sides of $6\text{ cm}$ and $10\text{ cm}$, a height of $4\text{ cm}$, and a solid length of $12\text{ cm}$. What is its volume?

1. $96{\text{ cm}}^3$
2. $384{\text{ cm}}^3$
3. $768{\text{ cm}}^3$
4. $1440{\text{ cm}}^3$

Show answer

$384{\text{ cm}}^3$. First, calculate the cross-sectional area of the trapezium: $A=\frac{1}{2}(6+10)\times 4=\frac{1}{2}(16)\times 4=32{\text{ cm}}^2$. Next, multiply the cross-sectional area by the length of the solid: Volume $V=32\times 12=384{\text{ cm}}^3$.

Slide 2 · Volumes of Special Prisms and TroughsModel · 10 min

Teach the full lesson on the projector

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