Mensuration

1. Let h be the height and a be the side of an equilateral triangle, then h = (3/2) a.
2. Circumference and area of a circle.
   If r is the radius of a circle, then
   (i) the circumference of the circle = 2r
   (ii) the area of the circle = r²
3. Area of a circular ring.
   If R and r are the radii of the bigger and smaller (concentric) circles, then
   area of the ring = (R² -r²).
4. Circumference and area of a sector of a circle.
   If r is the radius of the circle and the arc subtends an angle of n° at the center, then
   (i) the length of the arc = (n/360).2r = nr/180
   (ii) the area of the sector = (n/360).r²
5. Circumference and area of circumscribed and inscribed circles of an equilateral triangle.
   If R and r are the radii of the circumscribed and inscribed circles of the triangle, then
   (i) R = (2/3)h and r = (1/3)h
   (ii) the circumference of the circumscribed circle = 2R = (4/3)h
   (iii) the area of the circumscribed circle = R² = (4/9)h²
   (iv) the circumference of the inscribed circle = 2 r = (2/3)h
   (v) the area of the inscribed circle = r² = (1/9) h²
6. Circumference and area of circumscribed and inscribed circles of a regular hexagon.
   Let a be the side of a regular hexagon and R, r be the radii of the circumscribed and inscribed circles respectively of the hexagon, then
   (i) R = a and r = (3/2) a
   (ii) the circumference of the circumscribed circle = 2R = 2a
   (iii) the area of the circumscribed circle = R² = a²
   (iv) the circumference of the inscribed circle = 2r = 3 a
   (iv) the area of the inscribed circle = r² = (3/4)a².

Surface area and volume (of solids)

1. Solid Cylinder.
   Let r be the radius and h be height of a solid cylinder, then
   (i) curved (lateral) surface area = 2 rh
   (ii) total surface area = 2 r(h +r)
   (iii) volume = r²h
2. Hollow cylinder.
   Let R and r be the external and internal radii, and h be the height of a hollow cylinder, then
   (i) external curved surface area = 2Rh
   (ii) internal curved surface area = 2rh
   (iii) total surface area = 2(Rh +rh +R² -r²)
   (iv) volume of material = (R² -r²)h
3. Cone.
   Let r, h and l be the radius, height and slant height respectively of a cone, then
   (i) slant height = r² +h²
   (ii) curved (lateral) surface area = rl
   (iii) total surface area = r(l +r)
   (iv) volume = (1/3) r²h
4. Solid sphere.
   Let r be the radius of a solid sphere, then
   (i) surface area = 4r²
   (ii) volume = (4/3) r³
5. Spherical shell.
   Let R and r be the radii of the outer and inner spheres, then
   (i) thickness of the shell = R -r
   (ii) volume of material = (4/3) (R³ -r³)
6. Solid hemisphere.
   Let r be the radius of a hemisphere, then
   (i) curved (lateral) surface area = 2r²
   (ii) total surface area = 3 r²
   (iii) volume = (2/3) r³
7. Hemispherical shell.
   Let R and r be the radii of the outer and inner hemispheres, then
   (i) the thickness of the shell = R -r
   (ii) external curved surface area = 2 R²
   (iii) internal curved surface area = 2r²
   (iv) total surface area = (3R² +r²)
   (v) volume of material = (2/3)(R³ -r³)

Exercise

1. Find the length of the diameter of a circle whose circumference is 44 cm.
2. Find the circumference of a circle whose area is 144 cm².
3. How many times will the wheel of a car rotate in a journey of 88 km if it is known that the diameter of the wheel is 56 cm?
4. (a) Calculate the radius of a cycle wheel in cm which makes 70 revolutions in moving a distance of 154 m.
   (b) A bucket is raised from a well by means of a rope which is wound round a wheel of diameter 77 cm. Given that the bucket ascends in 1 minute 28 seconds with a uniform speed of 1·1 m/s, calculate the number of complete revolutions the wheel makes in raising the bucket. Take to be 22/7.
5. A road which is 7 m wide surrounds a circular park whose circumference is 352 m. Find the cost of paving the road at Rs 20 per m².
6. The sum of radii of two circles is 7 cm and difference of their circumferences is 8 cm. Find the circumferences of the two circles.
7. Two circles touch externally. The sum of their areas is 58 cm² and distance between their centers is 10cm. Find the radii of the two circles.
8. The diameters of the two circles are in the ratio 3 : 4 and the sum of the areas of the circles is equal to the area of a circle whose diameter is 36 cm. Find the radii of the circles.
9. A copper wire when bent in the form of an equilateral triangle has area 1213 cm². If the same wire is bent into the form of a circle, find the area enclosed by the wire.
10. (a) In the figure (i) given below, ABCD is a square inscribed in a circle of radius 7 cm. Calculate the area of the circle and the area of the shaded region.
    (b) In the figure (ii) given below, ABCD is a rectangle with sides AB = 42 cm and BC = 28 cm. Two quarter circles are drawn as shown in figure. Calculate the area of the shaded part.
    (i) (ii)
11. A rectangle with one side 4 cm is inscribed in a circle of radius 2·5 cm. Find the area of the rectangle.
12. (a) Find the area of the figure (i) given below in square cm correct to one decimal place. (Take = 22/7)
    (b) The figure (ii) given below represents a quadrant of a circle of radius 3·5 cm, center O. Given OD = 2 cm, calculate the area of the shaded region. (Take = 22/7).
    (i) (ii)
13. (a) In the figure (i) given below, two circles with centers A and B touch each other at the point C. If AC = 8 cm and AB = 3 cm, find the area of the shaded region. (Take = 22/7).
    (b) In the figure (ii) given below, ABCD is a square of side 14 cm. A, B, C and D are centers of circular arcs of radius 7 cm, find the area of the shaded region. (Take = 22/7).
    (i) (ii)
14. (a) The figure (i) given below represents a rectangular lawn 24 m by 18 m with all the four corners rounded off by quarter circles of radii 3 m as shown. Find the perimeter and the area of the lawn correct to 4 significant figures. Take = 3·142.
    (b) The figure (ii) given below shows a running track which comprises of two straight runs and semicircular tracks on the sides. AD = 21 m and DE = 3·5 m. Find the difference in the distance run by an athlete, if he runs along the outer edge or the inner edge.
    (i) (ii)
15. (a) In the figure (i) given below, semi circles are described on the sides of the right angled triangle PQR, right angled at P. If QR = 7 cm, calculate the area of the shaded region.
    (b) In the figure (ii) given below ABCD is a square inscribed in a circle. With B and D as centers, quarter circles are drawn. If the side of the square is 14 cm, find the area of the shaded region.
    [Hint. Required area = area of circle with radius 72 cm -area of square ABCD +
            2.(area of quarter circle with radius 14 cm -area of ABC).]
    (i) (ii)
16. (a) In the figure (i) given below, ABC is a right angled triangle, B = 90°, AB = 28 cm and BC = 21 cm. With AC as diameter and with BC as radius a quarter circle is drawn. Find the area of the shaded region correct to two decimal places. Take to be 22/7.
    (b) The boundary of the shaded region in the figure (ii) given below consists of four semi-circular arcs, the smallest two being equal. If the diameter of the larger is 14 cm and of the smallest is 3·5 cm, calculate the length of the boundary and the area of the shaded region. (Take = 22/7).
    (i) (ii)
17. (a) In the figure (i) given below, the boundary of the shaded region comprises of four semicircles and two quarter circles. If OA = OB = OC = OD = 7 cm and the straight lines AC, BD are perpendicular to each other, find the length of the boundary and the area of the shaded region.
    (b) In the figure (ii) given below, ABCD is a rectangle with sides AB = 19 cm and BC = 12 cm. The arcs AP, PQ and QR are quadrants of circles. Find the area of the shaded region and the length of the curved line.
    (i) (ii)
18. Find the radius of a circle if a 40° arc has length 4 cm. Hence find the area of the sector formed by this arc.
19. (a) In the figure (i) given below, OABC is a sector of a circle of radius 6 cm. Calculate the perimeter and the area of the shaded region. Give your answer correct to two decimal places.
    (b) The figure (ii), represents the wiper of a car with the dimensions given in the figure, find the area swept by the wiper and the perimeter of the shaded area.
    (i) (ii)
20. A circle is inscribed in a regular hexagon of side 23 cm. Find the circumference and the area of the inscribed circle.
21. Find the total surface area of an open cylindrical vessel of length 42 cm, and of external and internal diameters 20 cm and 6 cm respectively.
22. An open cylindrical vessel is made of a metal. The internal diameter is 7 cm, the internal depth is 10 cm and the metal is 5 mm thick. Calculate the capacity of the vessel and the volume of the metal.
23. In the figure below, a rectangular platform is given with a semi-circular end on one side. It is 23·5 m long from one end to the other. If the length of half the circumference is 22 m, find the cost of constructing the platform 2·5 m high at the rate of Rs 6 per cubic meter.
        
24. The uniform cross-section PQRSTU of an open tank is formed of a rectangular part QRTU and the two equal quarter circles PUQ and RTS. Given PS = 1·6 m, UT = 72 cm and the length of the tank PV = 2 m, calculate the volume of the water in cubic meters, which the tank can hold, when filled to capacity. (Take = 22/7)
       
25. The total surface area of a right circular cone of slant height 13 cm is 90 cm2. Calculate
    (i) its radius in cm. (ii) its volume in cm3.
    Give your answer to (ii) in terms of .
26. A circus tent has lower part cylindrical and upper part of conical shape. The radius of its base is 70 m. The total height of the tent is 30 m and that of the cylindrical part is 6 m. Find the area of the canvas required to make the tent.
27. The volume of a cone is the same as that of the cylinder whose height is 9 cm and diameter 40 cm. Find the radius of the base of the cone if its height is 108 cm.
28. The entire surface of a solid cone of base radius 3 cm and height 4 cm is equal to the entire surface of a solid right circular cylinder of diameter 4 cm. Find the ratio of (i) their curved surfaces. (ii) their volumes.
29. A hollow metallic cylindrical tube has an internal radius of 3 cm and height 21 cm. The thickness of the metal of the tube is 0·5 cm. The tube is melted and cast into a right circular cone of height 7 cm. Find the radius of the cone correct to one decimal place.
30. A cone of maximum volume is carved out of a block of wood of size 20 cm x 10 cm x 10 cm. Find the volume of the cone carved out correct to one decimal place. (Take = 3·1416).
31. From a solid cylinder whose height is 8 cm and radius is 6 cm, a conical cavity of height 8 cm and base radius 6 cm, is hollowed out. Find the volume of the remaining solid correct to 4 significant figures. ( = 3·1416).Also find the total surface area of the remaining solid.
32. From a solid cylinder of height 30 cm and radius 7 cm, a conical cavity of height 24 cm and of base radius 7 cm is drilled out. Find the volume and the total surface of the remaining solid. (Take = 22/7).
33. An open cylindrical vessel of internal diameter 7 cm and height 8 cm stands on a horizontal table. Inside this is placed a solid metallic right circular cone, the diameter of whose base is 3·5 cm and height 8 cm. Find the volume of the water required to fill the vessel. If the cone is replaced by another cone, whose height is 1·75 cm and the radius of whose base is 2 cm, find the drop in the water level.
34. A solid cone of height 12 cm and base radius 6 cm has the top 4 cm removed as shown in the adjoining below. Find the whole surface of the remaining solid.
       
35. The radii of the ends of a bucket 30 cm high are 21 cm and 7 cm. Find its capacity in liters and the amount of sheet required to make this bucket. (Take = 22/7).

Answers