Volumes just add up
Joining solids does not use up any space, so the volume is the sum of the volumes of the parts.
A cone on a hemisphere
A toy is a cone standing on a hemisphere of the same radius, 3 cm. The cone is 9 cm high. Part of the surface vanished when they were joined, but what about the space inside?
How does the volume of the whole toy compare with the volumes of its two parts?
What this lesson covers
The idea
The volume of a solid formed by joining basic solids is the sum of the volumes of its constituents, since no volume is lost in joining them.
A cone on a hemisphere
A toy is a cone standing on a hemisphere of the same radius, 3 cm. The cone is 9 cm high. Part of the surface vanished when they were joined, but what about the space inside?
How does the volume of the whole toy compare with the volumes of its two parts?
- It is the sum of the two volumes
- It is less than the sum, because some space is lost where they join
- It is more than the sum
Add the volumes
Change r and h with the dials. The working shows the volume of the hemisphere, of the cone, and of the whole toy. Press the button to take the pieces apart: look at what happens to the volume.
Nothing is lost
The volume of a combined solid is the sum of the volumes of its parts, because no space is lost when the parts are joined.
Surface area is different: the joined faces disappear, so we do not add total surface areas. But the pieces still fill the same space, whether they are apart or together. For the toy: hemisphere ⅔πr³ + cone ⅓πr²h.
With r = 3 cm and h = 9 cm (π = 3.14): 56.52 + 84.78 = 141.3 cm³.
Notes
The volume of a combined solid is the sum of the volumes of its parts, because no space is lost when they are joined.
Check yourself
A toy is a hemisphere surmounted by a cone. The radius of both is 2 cm and the cone is 2 cm high. Find the volume of the toy. (Take π = 3.14)
Answer: 25.12 cm³
Hemisphere = 2/3 × 3.14 × 8 = 16.75. Cone = 1/3 × 3.14 × 4 × 2 = 8.37. Toy = 25.12 cm³.
A shed is a cuboid with a half cylinder on top. The base of the shed is 7 m by 15 m and the cuboid is 8 m high. The half cylinder has diameter 7 m and length 15 m. How much air can the shed hold? (Take π = 22/7)
Answer: 1128.75 m³
Cuboid = 15 × 7 × 8 = 840. Half cylinder = ½ × 22/7 × 3.5 × 3.5 × 15 = 288.75. Total = 1128.75 m³.
A glass is a cylinder of inner diameter 5 cm and height 10 cm, but its bottom has a raised hemisphere that takes up space inside. Find the actual capacity of the glass. (Take π = 3.14)
Answer: 163.54 cm³
Cylinder = 3.14 × 2.5 × 2.5 × 10 = 196.25. Hemisphere = 2/3 × 3.14 × 2.5³ = 32.71. Capacity = 196.25 − 32.71 = 163.54 cm³.
Two solids are joined face to face. Which statement is true?
- The volume of the joined solid is the sum of the volumes; the surface area is less than the sum of the surface areas. — correct. Yes! The pieces still fill the same space, but the faces that touch are no longer on the outside.
- Both the volume and the surface area are the sum of the parts.. The surface area is not: the joined faces disappear from the outside.
- The volume is less than the sum, because some space is lost at the join.. No space is lost. The two solids together fill exactly the space that they filled apart.