Area and Volume Flashcards

Two concentric circles have radii 5 and 13. What fraction of the larger circle's area is occupied by the annular region between the two circles?

Answer: 144/169

The annulus area equals π(13² − 5²) = π(169 − 25) = 144π. The outer circle's area is π(13²) = 169π. The fraction is 144π ÷ 169π = 144/169. Note that 144 = 12² and 169 = 13², so this fraction does not simplify further.

A sphere of radius r is perfectly inscribed in a right circular cylinder, touching the top, bottom, and lateral surface. What fraction of the cylinder's volume does the sphere occupy?

Answer: 2/3

Since the sphere fits perfectly, the cylinder must have radius r and height 2r. Cylinder volume = πr²(2r) = 2πr³. Sphere volume = (4/3)πr³. The fraction is (4/3)πr³ ÷ 2πr³ = (4/3) ÷ 2 = 2/3. This is Archimedes' famous result.

A circle has radius 6. A chord subtends a central angle of 120°. What is the area of the circular segment (the region between the chord and the arc)?

Answer: 12π − 9√3

Sector area = (120/360)·π(6²) = (1/3)·36π = 12π. The triangle formed by the two radii and the chord has area = (1/2)(6)(6)sin(120°) = 18·(√3/2) = 9√3. Segment area = sector − triangle = 12π − 9√3.

A solid is formed by placing a hemisphere of radius 4 directly on top of a right circular cylinder of radius 4 and height 6, with the flat face of the hemisphere flush against the cylinder's top. What is the total exposed surface area of the solid?

Answer: 96π

The exposed surfaces are: (1) the circular bottom of the cylinder = π(4²) = 16π; (2) the lateral surface of the cylinder = 2π(4)(6) = 48π; (3) the curved surface of the hemisphere = 2π(4²) = 32π. The top circle of the cylinder and the flat base of the hemisphere are interior (they cancel each other). Total = 16π + 48π + 32π = 96π.

A right circular cone has height 9 cm and base radius 6 cm. A plane parallel to the base cuts the cone at a point 3 cm above the base, forming a frustum. What is the volume of the frustum?

Answer: 76π

The cut is 3 cm from the base, so 6 cm from the apex. By similar triangles, the small cone's radius = 6 × (6/9) = 4 cm. Volume of original cone = (1/3)π(6²)(9) = 108π. Volume of removed small cone = (1/3)π(4²)(6) = 32π. Frustum volume = 108π − 32π = 76π.

A square with side length 4 is inscribed in a circle. What is the area of one of the four circular segments lying between the circle and a side of the square?

Answer: 2π − 4

For a square of side 4 inscribed in a circle, the diagonal = 4√2, so the radius = 2√2. Circle area = π(2√2)² = 8π. Square area = 16. The four segments together have area 8π − 16. Each individual segment = (8π − 16)/4 = 2π − 4.