Chapter 12 · Kerala SSLC Class 10 Maths
Solids
Two ways to learn — a crisp formula reference, or a story that builds from folding a paper pyramid to comparing a sphere with the box that holds it.
Six scenes from folding a pyramid to a sphere in its enclosing cylinder.
How this works
Six scenes build from folding a paper pyramid all the way to comparing a sphere with the box that holds it. Read the story, then tap "The Maths" for the formal result.
Folding a Pyramid
Cut out a square in the middle with four equal isosceles triangles around it, fold up the triangles, and paste the edges together. What shape do you get?
Not a prism — a prism needs two equal bases and rectangles on the sides. This new solid has a square base, a single point on top, and triangles all around. It's called a pyramid.
The sides of the base polygon are base edges. The other sides of the triangles, running up to the top, are lateral edges. The topmost point is the apex.
Take a square pyramid with base edge 10 cm and lateral edge 13 cm. How much paper is needed to make it — its surface area?
Base area = 10² = 100 cm²
Since each triangular face is isosceles, its altitude bisects the 10 cm base — giving a right triangle with legs 5 cm and an unknown height, and hypotenuse 13 cm (the lateral edge).
triangle height = √(13² − 5²) = √(169−25) = √144 = 12 cm
area of one triangle = (1/2)(10)(12) = 60 cm²
This triangle height — the height of one lateral face once the pyramid is folded up — has its own name: the slant height of the pyramid.
Surface area = base + 4 triangles = 100 + 4(60) = 340 cm²
The Tent Problem
A tent is to be made in the shape of a square pyramid with base edge 6 m and height 4 m (the pyramid's actual height this time, not the lateral edge). How much canvas is needed?
A different right triangle connects height to slant height: from the apex, straight down to the centre of the base, then out to the midpoint of a base edge.
half base edge = 3 m, height = 4 m
slant height = √(3²+4²) = √(9+16) = √25 = 5 m
Four isosceles triangles, each with base 6 m and height (slant height) 5 m, make up the tent's surface.
area of one triangle = (1/2)(6)(5) = 15 m²
total canvas = 4 × 15 = 60 m²
Filling with Sand
Make a hollow square pyramid and a square prism with the SAME base and SAME height, both out of paper. Fill the pyramid with sand, then pour it into the prism. How much of the prism gets filled?
Exactly a third. Fill the pyramid two more times, and the prism is full — so the prism's volume is three times the pyramid's.
Volume of prism = base area × height
Volume of pyramid = (1/3) × base area × height
Try it: a square pyramid with base edge 10 cm and height 8 cm.
Volume = (1/3)(10²)(8) = (1/3)(100)(8) = 800/3 ≈ 266.7 cm³
A metal cube of edge 15 cm is melted and recast into a square pyramid of base edge 25 cm. Find the pyramid's height.
Volume of cube = 15³ = 3375 cm³ (unchanged after recasting)
Volume of pyramid = (1/3)(25²)(height) = 3375
(1/3)(625)(height) = 3375
height = 3375×3/625 = 16.2 cm
Rolling Up a Cone
Just as rolling up a rectangle makes a cylinder, rolling up a SECTOR of a circle makes a cone. What is the relationship between the sector and the finished cone?
The sector's radius becomes the cone's slant height
The sector's arc length becomes the circumference of the cone's base
A sector of central angle 45° is cut from a circle of radius 12 cm and rolled into a cone. What is the cone's base radius?
45° is 1/8 of the full 360°. Since arc length is proportional to central angle, this sector's arc is 1/8 of the full circle's circumference — and that arc becomes the base circle's circumference.
Since circumferences are proportional to radii, the base radius is also 1/8 of the original
base radius = (1/8)(12) = 1.5 cm
slant height = original radius = 12 cm
How do we make a cone of base radius 5 cm and slant height 15 cm?
The sector must come from a circle of radius 15 cm (matching the slant height). The base radius, 5, is 5/15=1/3 of that, so the arc — and hence the central angle — must also be 1/3 of the full circle.
central angle = 360° × (1/3) = 120°
The Conical Hat
To make a conical hat of base radius 8 cm and slant height 30 cm, how much paper is needed?
We need the AREA of the sector used to roll it up. The sector comes from a circle of radius 30 cm (the slant height). The base radius, 8 cm, is 8/30=4/15 of that radius — so the sector is 4/15 of the full circle, by area as well as by arc length.
sector area = (4/15) × π × 30² = (4/15) × 900π = 240π cm²
This is the curved surface area of the cone.
And exactly like the pyramid, filling a cone with sand and pouring it into a cylinder of the same base and height shows the cone holds exactly a third.
Volume of a cone = (1/3) × base area × height = (1/3)πr²h
For instance, a cone of base radius 4 cm and height 6 cm:
Volume = (1/3)π(16)(6) = 32π cm³
The Ball and Its Box
A sphere can't be cut open and spread flat the way a cone or pyramid can — its surface can't be flattened without stretching. But its surface area and volume are still known exactly: 4πr² and (4/3)πr³.
Take the smallest possible box for a ball — a cylinder that hugs the sphere exactly, with base radius r and height 2r (the sphere's diameter).
Cylinder surface area = 2πr(2r) + 2πr² = 4πr² + 2πr² = 6πr²
Sphere surface area = 4πr²
Ratio (cylinder : sphere) = 6πr² : 4πr² = 3 : 2
Cylinder volume = πr²(2r) = 2πr³
Sphere volume = (4/3)πr³
Ratio (cylinder : sphere) = 2 : (4/3) = 3 : 2
Both ratios come out exactly 3:2 — true for ANY radius r. The enclosing cylinder always has 1.5 times the surface area AND 1.5 times the volume of the sphere it wraps.
One more shape: a hemisphere — half a sphere. Its surface is the curved dome PLUS the flat circular cut. For a hemisphere of radius 12 cm:
Dome (half the sphere's surface) = (1/2)(4π×12²) = (1/2)(576π) = 288π cm²
Flat circle = π×12² = 144π cm²
Total surface area = 288π + 144π = 432π cm²
The Big Picture
Pyramids and cones share the same 1/3-of-a-prism volume rule; cones and cylinders share the same rolled-up construction idea. Master these seven moves and Chapter 12 is done.
Parts of a pyramid?
base edges, lateral edges, apex, slant height
Slant height from height?
slant² = height² + (half edge)²
Lateral edge from slant height?
lateral² = slant² + (half edge)²
Volume of a pyramid or cone?
(1/3) × base area × height
Cone from a sector?
sector radius=slant height l, arc=base circumference
Cone's curved surface area?
CSA = πrl
Sphere vs its enclosing cylinder?
cylinder:sphere = 3:2 (area & volume)
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