Surface Areas and Volumes

Why is the total surface area of a combined solid usually not the sum of the total surface areas of its parts?

When two solids are joined, the faces pressed together are hidden and no longer part of the outside surface. Only the exposed surfaces count, so a top made of a cone on a hemisphere has area CSA of hemisphere + CSA of cone.

Why is the volume of a combined solid simply the sum of the volumes of its parts?

Joining solids hides some surface, but no space inside either solid is lost. So, unlike surface area, the volume of the new solid is the total of the constituent volumes.

Curved surface area of a hemisphere

2πr², which is half of the sphere's surface area 4πr². If the flat circular face is exposed too, add its area πr².

Slant height of a cone

l = √(r² + h²), where r is the base radius and h the vertical height. The radius, height and slant height form a right triangle.

Curved surface area of a cone

πrl, where r is the base radius and l the slant height.

Curved surface area of a cylinder

2πrh, where r is the base radius and h the height. Add πr² for each circular end that is part of the surface.

Surface area of a cube

6 × (edge)², since a cube has six equal square faces.

Volume of a cylinder

πr²h, where r is the base radius and h the height. Half a cylinder has volume ½πr²h.

Volume of a cone

⅓πr²h, one-third of the volume of a cylinder with the same base radius and height.

Volume of a hemisphere

⅔πr³, half the volume of a sphere of the same radius.

A top is a cone on a hemisphere, 5 cm tall overall with diameter 3.5 cm. How do you find the area to colour?

The area is CSA of hemisphere + CSA of cone. The hemisphere's radius 1.75 cm is part of the height, so the cone is 5 − 1.75 = 3.25 cm tall and its slant height is about 3.7 cm. Then 2πr² + πrl ≈ 39.6 cm².

A hemisphere of diameter 4.2 cm sits on a cube of edge 5 cm. How do you find the block's total surface area?

Start with the cube's 150 cm², remove the circle covered by the hemisphere (πr²) and add the hemisphere's curved surface (2πr²). The net change is +πr², giving 150 + 13.86 = 163.86 cm².

A rocket is a cone of base radius 2.5 cm on a narrower cylinder of radius 1.5 cm. How do you work out the cone's painted area?

Paint the cone's curved surface plus the ring of its base left uncovered by the cylinder: πrl + πr² − πr′². With slant height 6.5 cm this is 3.14 × 20.25 ≈ 63.585 cm².

A bird-bath is a cylinder of height 1.45 m and radius 30 cm with a hemispherical hollow at the top. How do you find its total surface area?

Only the cylinder's curved surface and the inside of the hemispherical hollow are counted: 2πrh + 2πr² = 2πr(h + r). With r = 30 cm and h = 145 cm this is 33000 cm², or 3.3 m².

A shed is a 15 m × 7 m × 8 m cuboid topped by a half cylinder. How do you find the air it holds with machinery and workers inside?

Add the cuboid's volume to half the cylinder's (diameter 7 m, length 15 m) to get 1128.75 m³. Then subtract the space taken by the machinery (300 m³) and the workers (1.6 m³), leaving 827.15 m³.

A cylindrical glass has a raised hemispherical bottom. How do you find its actual capacity?

Its apparent capacity is the cylinder's volume, πr²h, here 196.25 cm³. The raised hemisphere takes up ⅔πr³, about 32.71 cm³, so the actual capacity is the difference, 163.54 cm³.

A toy is a cone of height 2 cm on a hemisphere of radius 2 cm. How do you find the space between it and a cylinder that just encloses it?

Toy volume = ⅔πr³ + ⅓πr²h = 25.12 cm³. The enclosing cylinder has radius 2 cm and height 2 + 2 = 4 cm, so its volume is 3.14 × 4 × 4 = 50.24 cm³. The difference is 25.12 cm³.

Two equal cubes of edge a are joined face to face into a cuboid. Why is its surface area 10a² rather than 12a²?

Each cube has 6 faces of area a², 12 in all, but the two faces pressed together are hidden inside the cuboid. That leaves 10 exposed faces, so the surface area is 10a².

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