Flashcards · 14 cards

Solid mensuration

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Rule
The volume of a prism?
Answer
Area of the cross-section × length.
From the lesson: Prisms and cylinders
Rule
The volume and curved surface area of a cylinder?
Answer
$V = \pi r^2 h$ and curved surface $= 2\pi rh$ . Add $\pi r^2$ for each closed end.
From the lesson: Prisms and cylinders
Know it
Surface area: how many circular ends for a closed can, a tank open at the top, and a pipe open at both ends?
Answer
Closed can: 2. Open-topped tank: 1. Pipe open at both ends: none.
From the lesson: Prisms and cylinders
Rule
The volume of metal in a pipe?
Answer
The ring between the two circles, times the length: $\pi(R^2 - r^2)h$ .
The end of a pipeThe metal is the ring between the two circles, all the way along
From the lesson: Prisms and cylinders
Rule
The longest diagonal of a cuboid $l \times b \times h$ ?
Answer
Pythagoras twice: $\sqrt{l^2 + b^2 + h^2}$ .
Pythagoras twiceFloor diagonal first, then up to the opposite top corner
From the lesson: Prisms and cylinders
Rule
The volume of a cone, and of a pyramid?
Answer
$\frac13 \times$ base area $\times$ height. For a cone, $\frac13\pi r^2 h$ .
From the lesson: Cones and pyramids
Rule
In a cone, how are $h$ , $r$ and the slant height $l$ related? Which does each formula use?
Answer
$l^2 = h^2 + r^2$ . The volume uses $h$ ; the curved surface $\pi rl$ uses $l$ .
Height, radius, slant heightA right-angled triangle inside every cone
From the lesson: Cones and pyramids
Rule
A sector is bent into a cone. What becomes what?
Answer
The sector's radius becomes the slant height; its arc becomes the circumference of the base.
Bending a sectorThe radius becomes the slant height; the arc becomes the base circle
From the lesson: Cones and pyramids
Rule
The volume and surface area of a sphere?
Answer
$V = \frac43\pi r^3$ and $S = 4\pi r^2$ . A hemisphere holds half: $\frac23\pi r^3$ .
From the lesson: Spheres, composite solids and capacity
Rule
How do you find the volume of a frustum?
Answer
Put the tip back: the volume of the big cone minus the small cone cut off.
A frustumPut the tip back to find H; the slant height comes from the small triangle
From the lesson: Spheres, composite solids and capacity
Rule
Similar solids with lengths in the ratio $k$ : how do areas and volumes compare?
Answer
Areas × $k^2$ , volumes × $k^3$ .
Similar solidsLengths × k, areas × k², volumes × k³
From the lesson: Spheres, composite solids and capacity
Know it
How many litres are in 1 m³? How many cm³ in a litre?
Answer
1 m³ = 1000 litres, and 1 litre = 1000 cm³. Change to one unit before you divide.
From the lesson: Spheres, composite solids and capacity
Which method?
WAEC 2022 · Paper 2 · Q8 (b)
A sector of angle $220^\circ$ is removed from a thin circular metal sheet of radius $63\text{ cm}$ . It is then folded with the straight edges meeting to form a right circular cone. Calculate, correct to one decimal place, the: (i) base radius; (ii) volume, of the cone. $\left[\text{Take }\pi = \frac{22}{7}\right]$
How do you find the cone's radius?
Answer
The sector's arc becomes the base circumference: $\frac{220}{360} \times 2\pi \times 63 = 2\pi r$ , so $r = \frac{220}{360} \times 63$ . The 63 cm becomes the slant height.
From the lesson: Cones and pyramids Try the question
Which method?
WAEC 2020 · Paper 2 · Q7 (a)
The diagram shows a wooden structure in the form of a cone, mounted on a hemispherical base. The vertical height of the cone is $48\text{ m}$ and the base radius is $14\text{ m}$ . Calculate, correct to three significant figures, the surface area of the structure. $\left[\text{Take }\pi = \frac{22}{7}\right]$
Which surfaces count?
Answer
Only the outside: the cone's curved surface ( $\pi rl$ , with $l$ from Pythagoras) and the hemisphere's curved surface ( $2\pi r^2$ ). The circle where they join is hidden.
From the lesson: Spheres, composite solids and capacity Try the question