Plane Mensuration: It deals with the sides, perimeters and areas of plane figures of different shapes.
Solid Mensuration: It deals with the areas and volumes of solid objects
Rectangle: A rectangle is a plane, whose opposite sides are equal and diagonals are equal. Each angle is equal to 90˚.
2. Area of rectangle = length x breadth = l x b = 4 x 3
= 12 \(m^{2}\)
3. Length of a rectangle : \(\frac{area}{breadth}\) = \(\frac{A}{b}\) = \(\frac{12}{3}\) = 4 m
Or, [\(\frac{perimeter}{2}\) – breadth] = (\(\frac{14}{2}\) – 3) = 4m
Breath of a rectangle : \(\frac{area}{length}\) = \(\frac{A}{l}\) = \(\frac{12}{4}\) = 3 m
Or, [\(\frac{perimeter}{2}\) – length] = (\(\frac{14}{2}\) – 4) = 3 m
Parallelogram: It is a rectilinear figure with opposite sides parallel.
Rhombus: It is a simple quadrilateral in which all sides have same length. A rhombus is often called as ‘Diamond’.
Trapezium: It is a quadrilateral with one of pair of sides parallel.
Triangle: It is a polygon with three edges and three vertices.
Equilateral triangle: It is defined as a polygon with three equal sides. Each and every angle in it is 60 degrees.
Cube: A cube is a three dimensional figure and has six square faces which meet each other at right angles. It has eight vertices and twelve edges.
2. Side of a cube = \(\sqrt[3]{Volume}\)
= \(\sqrt[3]{1728}\)
= 12 cm
3. Diagonal of cube = \(\sqrt{3}\) x (Side) = \(\sqrt{3}\) x 12 = 12 \(\sqrt{3}\) cm
4. Total surface area of a cube = 6 x \((Side)^{2}\) = 6 x \((12)^{2}\) = 864 sq.cm
Cuboid (Rectangular parallelopiped): A solid body having six rectangular faces, is called cuboid. (or) A parallelopiped whose faces are rectangles is called rectangular parallelopiped or cuboid.
2. Volume of a cuboid = (length × breadth × height) = lbh
= 12 × 8 × 6 = 576 cuboic cm
3. Diagonal of a cuboid = \(\sqrt{l^{2} + b^{2} + h^{2}}\) = \(\sqrt{12^{2} + 8^{2} + 6^{2}}\)
= \(\sqrt{144 + 64 + 36}\) = \(\sqrt{244}\) = \(2\sqrt{61}\) cm.
4. Length of cuboid = \(\frac{Volume}{Breadth \times Height}\) = \(\frac{v}{b \times h}\)
5. Breadth of cuboid = \(\frac{Volume}{Length \times Height}\) = \(\frac{v}{l \times h}\)
6. Height of cuboid = \(\frac{Volume}{length \times breadth}\) = \(\frac{v}{l \times b}\)
Cylinder: A solid geometrical figure with straight parallel sides and a circular or oval cross section is called cylinder.
2. Total surface area = area of circular ends + curved surface area
= \(2 \pi r^{2}\) + \(2 \pi r(r + h) \) sq. unit
= 2 x \(\frac{22}{7}\) x 7(15 + 7)
= 2 x 22 x 22
= 968 sq. cm.
4. Volume of a hollow cylinder = \(\pi R^{2}h – \pi r^{2}h\)
= (\(\pi h (R^{2} – r^{2})\)) = \( \pi h(R + r)(R – r)\)
= \( \pi \times height \times (sum of radii)(difference of radii)\)
Here R, r are outer and inner radii respectively and h is the height.
Cone: A solid (3-dimensional) object with a circular flat base joined to a curved side that ends in an apex point is called cone.
2. Curved surface area = \(\frac{1}{2}\) x (perimeter of base) x slant height
= \(\frac{1}{2}\) x \(2 \pi r\) x l = \(\pi r l\) sq. unit
= \(\frac{22}{7}\) x 35 x 37 = 4070 sq. cm
3. Total surface area S = area of circular base + curved surface area = \((\pi r^{2} + \pi r l)\) = \(\pi r (r + l)\) sq. unit
= \(\frac{22}{7}\) x 35 (37 + 35) = 7920 sq. cm
4. Volume of cone = \(\frac{1}{3}\) (area of base) x height
= \(\frac{1}{3} (\pi r^{2})\) x h = \(\frac{1}{3} \pi r^{2} h\) cubic unit
= \(\frac{1}{3}\) x \(\frac{22}{7}\) x 35 x 35 x 12
= 15400 cubic cm
Frustum of Cone:
6. Lateral surface = \(\pi l (R + r)\)
Where \(l^{2}\) = \(h^{2}\) + \((R – r)^{2}\)
7. Total surface area = \(\pi [R^{2} + r^{2} + l(R + r)]\)
R, r be the radius of base and top the frustum
ABB ‘A’ h and l be the vertical height and slant
Height respectively.
Sphere: A 3-dimensional object shaped like a ball is called sphere and every point on the surface is the same distance from the centre.
2. Radius of sphere = \(\sqrt{ \frac{surface area}{4 \pi}}\) = \(\sqrt{ \frac{1386 \times 7}{4 \times 22}}\) = 10.5 cm
3. Diameter of sphere = \(\sqrt{ \frac{surface}{4 \pi}}\) = \(\sqrt{ \frac{surface}{4 \pi}}\) = \(\sqrt{ \frac{1386 /times 7}{22}}\) = 21 cm
4. Volume of sphere V = \(\frac{4}{3} \pi r^{3}\) = \(\frac{4}{3} \pi (\frac{d}{2})^{3}\) = \(\frac{1}{6} \pi d^{3}\)
= \(\frac{1}{6}\) x \(\frac{22}{7}\) x 21 x 21 x 21 = 4831 cubic cm
5. Radius of sphere = \(\sqrt{ \frac{3}{4 \pi} \times volume of shpere}\)
6. Diameter = \(\sqrt[3]{ \frac{6 \times v}{\pi}}\)
7. Volume of spherical ring = \(\frac{4}{3} \pi (R^{3} – r^{3}\)
Hemisphere: In geometry, hemisphere is an exact half of sphere.
8. Curved surface of hemisphere = 2 \(\pi r^{2}\)
9. Volume of hemisphere = \(\frac{2}{3} \pi r^{3}\)
10. Total surface area of hemisphere = \(3 \pi r^{2}\)
Circle: It is defined as, if a straight line is bent until its ends join. It is a simple closed curve and the distance between any of the points and centre of circle is called radius.
2. Radius of a circle = \(\frac{circumference}{2 \pi}\) = \(\frac{264 \times 7}{2 \times 22}\) = 42 cm
3. Area of a circle = \(\pi \times r^{2}\) = \(\frac{22}{7}\) x \(42^{2}\) = \(\frac{22}{7}\) x 42 x 42 = 5544 \(cm^{2}\)
4. Radius of a circle = \(\sqrt{\frac{area}{\pi}}\)
= \(\sqrt{\frac{5544}{22} \times 7}\) = \(\sqrt{1764}\) = 42cm
Semi-circle: It is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points.
5. Area of a semi circle = \(\frac{1}{2} \pi r^{2}\) = \(\frac{1}{8} \pi d^{2}\)
= \(\frac{1}{2}\) x \(\frac{22}{7}\) x \(42^{2}\) = \(2772 cm^{2}\)
6. Circumference of semi circle = \(\frac{22}{7}\) x 42 = 132 cm
7. Perimeter of semi circle = \((\pi r + 2r)\) = \((\pi + 2)r\) = \((\pi + 2)\frac{d}{2}\)
8. Area of sector OAB = \(\frac{x}{360}\) x \(\pi r^{2}\)
(x being the central angle)
= \(\frac{30^{0}}{360^{0}}\) x \(\frac{22}{7}\) x 3.5 x 3.5 = 3.21 sq. m.
9. Central angle by arc AB = \(360^{0}\) x \(\frac{area of OAB}{are of circle}\)
= \(360^{0}\) x \(\frac{3.21}{\frac{22}{7} \times 3.5 x 3.5}\) = \(\frac{360 \times 321}{22 \times 35 \times 5}\) = \(30^{0}\)(approx)
10. Radius of circle = \(\sqrt{\frac{360^{0}}{central \ angle \ by \ arc} \times \frac{area \ of \ OAB}{\pi}}\)
= \(\sqrt{\frac{360^{0}}{30^{0}} \times \frac{3.21}{\frac{22}{7}}}\) = \(\sqrt{\frac{134.82}{11}}\) = \(\sqrt{12.23}\) = 3.5 m.
11. Area of ring = difference of the area of two circle
= \(\pi R^{2}\) – \(\pi r^{2}\) = (\(R^{2} – r^{2}\))
= \(\pi\)(R + r)(R – r)
= (sum of radius)(diff. of radius)
= \(\frac{22}{7}\) x (4 + 3)(4 – 3) = \(\frac{22}{7}\) x 7 x 1
= 22 sq. cm.
2. Rectangle:
Area = Length x Breadth
Perimeter = 2(Length x Breadth)
Diagonal = \(\sqrt{l^2 + b^2}\)
3. Parallelogram:
Area = Base x Height
Perimeter = 2(Length x Breadth)
Base = \(\frac{Area}{Height}\)
4. Rhombus:
Area = \(\frac{1}{2}* d_{1} d_{2}\)
Where, \(d_{1} \ and \ d_{2}\) are diagonals.
Perimeter = 4 x side
Diagonal = \(\frac{2 * area}{other diagonal}\)
5. Trapezium:
Area = \(\frac{1}{2}(a + b) * h\)
Perimeter = Sum of the sides
6. Triangle:
Area = \(\frac{1}{2} * Base * Height\) (or) \(\sqrt{s(s – a)(s – b)(s – c)}\)
Where a, b, c are sides of triangle and s = \(\frac{a + b + c}{2}\)
Perimeter = Sum of the sides
7. Equilateral Triangle:
Area = \(\frac{\sqrt{3}}{2}(side)^2\)
Perimeter = 3(Side)
8. Cube:
Let each edge of a cube be of length \(a\). Then,
Volume = \(a^3\) cubic units
Surface area = 6\(a^2\) sq. units
Diagonal = \(\sqrt{3}a\)units
9. Cuboid:
Let l – length, b – breadth, h – height. Then,
Volume = (l x b x h) cubic units
Surface area = 2(lb + bh + lh) sq. units
Diagonal = \(\sqrt{l^2 + b^2 + h^2}\) units
10. Cylinder:
Let radius of base = r and height( or length) = h. Then,
Volume = \(\pi r^2 h \) cubic units
Curved Surface area = 2\(\pi r h\) sq. units
Total surface area = 2(\(\pi rh + 2\pi r^2\)) = 2\(\pi r(h + r)\)sq. units
11. Cone:
Let radius of base = r and height = h. Then,
Slant height, l = \(\sqrt{h^2 + r^2}\) units
Volume = \(\frac{1}{3}\pi r^2 h\) cubic units
Curved surface area = \(\pi r l\) sq. units
Total surface area = \(\pi r l + \pi r^2\) sq. units
12. Sphere:
Let radius of the sphere be r. Then,
Volume = (\(\frac{4}{3} \pi r^3\)) cubic units
Surface area = \(\pi r^2\) sq. units
13. Hemisphere:
Let the radius of a hemisphere be \(r\).
Volume = \(\frac{2}{3} \pi r^3\) cubic units.
Curved surface area = 2\(\pi r^2\) sq. units.
Total surface area = 3\(\pi r^2 \) sq. units.
14. Prism:
Volume of right prism = (Area of the base * height)cu. units
Lateral surface area of a right prism = (perimeter of the base * height)sq. units
Total surface area of a right prism = lateral area + 2(area of one base)sq. units
15. Circle:
Area = \(\pi r^2\)
where, r = radius
Circumference = \(2 \pi r \)
16. Semi-circle:
Area = \(\frac{1}{2} \pi r^2\)
Circumference = \(\pi r + 2r\)
17. Sector:
Area = \(\frac{\theta}{360} * \pi r^2\)
Circumference = l + 2r[l = \(\frac{\theta}{360} * 2\pi r\)]
where l = length of arc.
2. Expenditure incurred in cultivating a square field at the rate of Rs. 170 per hectare is Rs. 680. What would be the cost of fencing the field at the rate of Rs. 3 per metre.
Solution:
3. A man walking at the rate of 6 km per hour crosses a square field diagonally in 9 seconds. Find the area of the field ?
Solution:
4. Find the diagonal of a cuboid whose dimensions are 22 cm, 12 cm, and 7.5 cm?
Solution:
5. The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. If the total surface area is 616\({cm}^2\), then find the volume of the cylinder?
Solution: