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Prisms and pyramids
This test develops geometric reasoning and calculations with prisms and pyramids. Students find volume, lateral and total surface area, altitude and edges and analyse diagonal cross-sections.
Questions: 10Estimated time: 23 minutes
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Question 1out of 10
What does a cross-section of a solid mean?
A section is the intersection of a solid with a plane. Identify its shape and dimensions before calculating. An axial section plane passes through the rotation axis; section area is not the total surface area.
Question 2out of 10
The base edge of a regular square pyramid is 20 cm long, and the height of the pyramid is 24 cm. Calculate the length of the slant height (apothem).
The slant height, vertical height h and half the base side a/2 form a right triangle. Slant height l = √(h² + (a/2)²).
Question 3out of 10
A right prism has rectangular base sides 24 cm and 45 cm and height 7 cm. Find its diagonal-section area.
The diagonal section of a right prism is a rectangle. Its width is the base diagonal √(a² + b²) and height h. Section area = h√(a² + b²).
Question 4out of 10
How are the volumes of a prism and pyramid with equal base and height related?
Prism volume V = base area × height. Pyramid volume V = base area × height / 3. A pyramid has one third of the volume of a prism with the same base and height. Use perpendicular height and cm³.
Question 5out of 10
A right prism has base area 150 cm² and height 13 cm. Find its volume.
Prism volume V = base area × height. Pyramid volume V = base area × height / 3. A pyramid has one third of the volume of a prism with the same base and height. Use perpendicular height and cm³.
Question 6out of 10
A rectangular prism measures 4 cm, 7 cm and 13 cm. Find total surface area.
A rectangular prism has three pairs of equal rectangular faces: S = 2(ab + ah + bh). Surface area uses cm², not cm³.
Question 7out of 10
A pyramid has base area 6 cm² and height 12 cm. Find its volume.
Prism volume V = base area × height. Pyramid volume V = base area × height / 3. A pyramid has one third of the volume of a prism with the same base and height. Use perpendicular height and cm³.
Question 8out of 10
A right prism has rectangular base sides 8 cm and 15 cm and height 12 cm. Find its diagonal-section area.
The diagonal section of a right prism is a rectangle. Its width is the base diagonal √(a² + b²) and height h. Section area = h√(a² + b²).
Question 9out of 10
The base edge of a regular square pyramid is 8 cm long, and the slant height (apothem) is 5 cm. Calculate the total surface area of the pyramid.
Total area is the square base plus four triangular faces: S = a² + 4 × al/2 = a² + 2al. l is the face slant height, not the pyramid vertical height.
Question 10out of 10
The base edge of a regular square pyramid is 7 cm long, and the slant height (apothem) is 9 cm. Calculate the total surface area of the pyramid.
Total area is the square base plus four triangular faces: S = a² + 4 × al/2 = a² + 2al. l is the face slant height, not the pyramid vertical height.
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Sample questions
- A right prism has base area ___ cm² and height ___ cm. Find its volume.
- A rectangular prism measures ___ cm, ___ cm and ___ cm. Find total surface area.
- A right prism has rectangular base sides ___ cm and ___ cm and height ___ cm. Find its diagonal-section area.
- A pyramid has base area ___ cm² and height ___ cm. Find its volume.
- The base edge of a regular square pyramid is ___ cm long, and the height of the pyramid is ___ cm. Calculate the length of the slant height (apothem).
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