S.S.1 THIRD TERM MATHEMATICS WEEK 1
Topic: MENSURATION
CUBES
A cube is a solid of uniform cross-section. It is formed by squares and has 8 vertices. An example is processed cubed sugar.
The length of a side of a cube is ‘e’ which is the length of all sides since a cube is formed with squares.
TOTAL SURFACE AREA OF A CUBE
A cube has 6 faces. The surface area of each side = e2 as each side is a square.
Therefore, Total surface area of a cube (all 6 sides) = 6e2
The surface area of a cube is gotten by the formula
Surface area= 6e2 sq. units
VOLUME OF A CUBE
In a cube all sides are equal. Length=e, height=e and width=e
Therefore, Volume of a cube= length× width× height
Volume of a cube= e3 cubic units
CYLINDERS
A Cylinder is a uniform circular cross-section. Examples of cylinders are unsharpened pencils like HB or 2B pencils, garden rollers, tins of milk or tomato et cetera
TOTAL SURFACE AREA OF A CYLINDER
There are two types of cylinders;
(1) A closed cylinder and
(2) An open cylinder
TOTAL SURFACE AREA OF A CLOSED CYLINDER
The total surface area of a closed cylinder consists of a sum of the areas of (i) the curve surface and (ii) The two circular end faces.
The curved surface when opened out is a rectangle. This rectangle has length equal to the length of the
Cylinder and the width are equal to the circumference of the circular end face.
Area of curved surface of a cylinder = area of rectangle of dimensions length (L) and width
(Circumference of base)
= 2πrl
Area of the two circular end faces= twice the area of one circular face
= πr2
Hence the total surface area of the closed cylinder
= 2πrl + 2πr2 sq. units
TOTAL SURFACE AREA OF AN OPEN CYLINDER
The total surface area of an open cylinder is the area of the curved surface which is the area of the rectangle the cylinder forms when spread.
Sometimes we are given a thick hollow cylinder. The total surface area is the sum of;
(i) the area of the external curved surface
(ii) the area of the internal curved surface and
(iii) the area of the end annular faces which will be shaded.
VOLUME OF A CYLINDER
A right circular cylinder is a solid of uniform cross-section. If a paper is wrapped round a cylinder, on opening it, a rectangle will be found.
Thus if the height of a cylinder
= h units
And the base radius
= r units
Then the volume of a cylinder
= Area of base by height
= πr2h cubic units
TRIANGULAR PRISMS
A prism is a solid with uniform cross-section of a shape of a triangle or a trapezium or any other polygon.
TOTAL SURFACE AREA OF A TRIANGULAR PRISM
In the case of a triangular shaped prism, the total surface area is the sum of the surface areas of the five faces that make up the prism.
VOLUME OF A TRIANGULAR PRISM
The volume of a prism is the area of its cross-section multiplied by the distance between the end faces. Examples are funnel, Chinese hat, cut periwinkle shell et cetera.
CONES
A cone is a figure with circular base and sides slanting to a common point or vertex. There are two types of cones (i) a right circular cone, where the line joining the vertex is symmetrical and perpendicular to the base of the cone and (ii) the non-right circular cone but this isn’t in the syllabus.
SURFACE AREA OF A CONE
Since a cone is formed from a sector of circle, then the surface area of a cone is equal to the area of the sector that formed it. Let L be the radius of the sector, then L becomes the slant height of the cone. If r is the radius of the base of the cone, then the length of arc of the sector is equal to 2pi.r which equals the circumference of the base of the cone.
If the sector subtends an angle- which it always does- then the area of the sector will be equal to
= θ /360 ×πl2 = curved surface of a cone
But 2πr = length of arc of a sector.
Therefore, 2πr = θ /360 x 2πl
Finally, surface area of a cone = πrl
TOTAL SURFACE AREA OF A CONE
The total surface area of cone is the sum of (i) the curved surface area πrl sq. units and (ii) the area of the base of the cone πr2 sq. units.
Therefore, total surface area of a cone = πr2 + πrl
