Surface Areas and Volumes
This chapter covers finding the surface area and volume of combinations of basic solids like cubes, cuboids, cylinders, cones, and spheres, which commonly appear together in real-life objects.
Cylinder: Curved Surface Area = 2(pi)rh, Volume = (pi)r^2h. Cone: Curved Surface Area = (pi)rl (l = slant height), Volume = (1/3)(pi)r^2h. Sphere: Surface Area = 4(pi)r^2, Volume = (4/3)(pi)r^3. Hemisphere: Curved Surface Area = 2(pi)r^2, Volume = (2/3)(pi)r^3.
The slant height (l) of a right circular cone is the distance from the apex to any point on the circumference of the base, related to the radius (r) and height (h) by: l^2 = r^2 + h^2 (Pythagoras theorem).
Q. A solid is in the shape of a cone standing on a hemisphere, with both having the same radius of 1 cm, and the height of the cone is equal to its radius. Find the volume of the solid.
Volume of cone = (1/3)(pi)r^2h = (1/3)(pi)(1)^2(1) = pi/3 cm^3. Volume of hemisphere = (2/3)(pi)r^3 = (2/3)(pi)(1)^3 = 2pi/3 cm^3. Total volume = pi/3 + 2pi/3 = 3pi/3 = pi cm^3 (approximately 3.14 cm^3).
When two solids are joined together, the total surface area is NOT simply the sum of their individual total surface areas, since the surfaces where they join are no longer exposed and must be excluded.
Q. A cylindrical tub of radius 12 cm contains water up to a depth of 20 cm. A spherical ball is dropped and fully submerged, raising the water level by 6.75 cm. Find the radius of the ball.
Volume of water raised = volume of the ball. (pi)(12)^2(6.75) = (4/3)(pi)r^3. 144 x 6.75 = (4/3)r^3, so 972 = (4/3)r^3, giving r^3 = 729, so r = 9 cm.
The volume of a sphere with radius r is:
Correct answer: (B) (4/3)(pi)r^3
When a solid is converted from one shape to another by melting, the _______ of the material remains the same.
Volume
The total surface area of a combination of two solids is always the sum of their individual total surface areas.
False — the surfaces at the point where two solids are joined become internal and are not part of the combined solid's exposed surface, so they must be subtracted when calculating total surface area.