Geometric Area and Volume Flashcards

A sphere is inscribed inside a cube so that it touches all six faces. If the volume of the cube is 216 cm³, what is the surface area of the sphere?

Answer: 36π cm²

The cube has volume 216 cm³, so its side length is ∛216 = 6 cm. A sphere inscribed in a cube has a diameter equal to the cube's side length, giving diameter = 6 cm and radius = 3 cm. Surface area of a sphere = 4πr² = 4π(3²) = 36π cm².

A composite solid consists of a cylinder with radius 5 m and height 8 m, with a cone of the same base radius removed from its top. The cone has a height of 6 m. What is the volume of the remaining solid?

Answer: 150π m³

Volume of the full cylinder = πr²h = π(5²)(8) = 200π m³. Volume of the removed cone = (1/3)πr²h = (1/3)π(25)(6) = 50π m³. Remaining volume = 200π − 50π = 150π m³.

A rectangular box has a space diagonal of 14 cm, a length of 12 cm, and a width of 4 cm. What is the volume of the box?

Answer: 288 cm³

The space diagonal satisfies d² = l² + w² + h². Substituting: 14² = 12² + 4² + h², so 196 = 144 + 16 + h², giving h² = 36 and h = 6 cm. Volume = 12 × 4 × 6 = 288 cm³.

Two similar pyramids have surface areas in the ratio 16:25. If the volume of the smaller pyramid is 128 cm³, what is the volume of the larger pyramid?

Answer: 250 cm³

For similar solids, the ratio of surface areas equals the square of the linear scale factor. So the linear ratio is √(16/25) = 4/5. The ratio of volumes equals the cube of the linear ratio: (4/5)³ = 64/125. Setting up the proportion: 128/V = 64/125, so V = 128 × (125/64) = 250 cm³.

A sector of a circle has a central angle of 120° and an arc length of 8π cm. What is the area of the sector?

Answer: 48π cm²

Arc length = (θ/360°) × 2πr. So 8π = (120/360) × 2πr = (1/3)(2πr). Solving: 8π = 2πr/3, which gives r = 12 cm. Area of sector = (θ/360°) × πr² = (120/360) × π(12²) = (1/3)(144π) = 48π cm².

A hollow cylindrical pipe has an outer radius of 7 cm, an inner radius of 5 cm, and a height of 10 cm. What is the volume of material that makes up the pipe walls?

Answer: 240π cm³

The volume of material equals the volume of the outer cylinder minus the volume of the inner hollow: V = π(R² − r²)h = π(7² − 5²)(10) = π(49 − 25)(10) = π(24)(10) = 240π cm³.