Compound Shapes and Parts of Shapes (Copy)
1.
A rectangle 12 cm by 8 cm is attached to a semicircle of diameter 12 cm.
Find the perimeter of the shape in terms of π.
2.
Find the area of the shape in Q1 in terms of π.
3.
A square of side 10 cm has a semicircle of diameter 10 cm on top.
Find the total area in terms of π.
4.
A rectangular garden 20 m by 15 m has a semicircular flower bed of diameter 15 m on one side.
Find the area of the garden in terms of π.
5.
A circle of radius 7 cm is cut into a semicircle.
Find the perimeter of the semicircle in terms of π.
6.
A circle of radius 14 cm is cut into a quadrant.
Find its area in terms of π.
7.
A rectangle 16 cm by 10 cm has four semicircles drawn outward on each side.
Find the perimeter in terms of π.
8.
A square of side 14 cm has four semicircles attached to its sides.
Find the total area in terms of π.
9.
A circular field of radius 21 m has a circular path of width 3 m around it.
Find the area of the path in terms of π.
10.
A circular ring has outer radius 10 cm and inner radius 6 cm.
Find its area in terms of π.
11.
A hollow cylinder has outer radius 10 cm, inner radius 7 cm, and height 15 cm.
Find its volume in terms of π.
12.
Find the curved surface area of the hollow cylinder in Q11.
13.
A cone has base radius 7 cm and height 24 cm. The top portion of height 8 cm is cut off to form a frustum.
Find the volume of the frustum in terms of π.
14.
Find the slant height of the frustum in Q13.
15.
A cone of height 30 cm and base radius 14 cm is cut into two parts by a plane parallel to the base at half the height.
Find the volume of the smaller cone in terms of π.
16.
Find the volume of the frustum left in Q15.
17.
A solid consists of a hemisphere of radius 7 cm fixed on top of a cylinder of radius 7 cm and height 10 cm.
Find its total volume in terms of π.
18.
Find the curved surface area of the solid in Q17 in terms of π.
19.
A cone of radius 9 cm and height 12 cm is placed on top of a hemisphere of radius 9 cm.
Find the total height of the solid and its volume in terms of π.
20.
A solid consists of a cone of height 24 cm and base radius 7 cm on top of a cylinder of height 30 cm and radius 7 cm.
Find the volume of the solid in terms of π.
21.
A solid is made of a cuboid of dimensions 8 cm × 8 cm × 10 cm, with a hemisphere of radius 4 cm fixed on top.
Find its total volume in terms of π.
22.
A cuboid of size 6 cm × 6 cm × 8 cm has a cylindrical hole of radius 2 cm drilled through its height.
Find the remaining volume in terms of π.
23.
A cube of side 8 cm has a cylindrical hole of radius 3 cm drilled through it. Height of hole = 8 cm.
Find the volume of the remaining solid in terms of π.
24.
A sphere of radius 6 cm is cut into two hemispheres.
Find the total surface area of the two hemispheres in terms of π.
25.
A frustum of a cone has height 20 cm, radii 12 cm and 8 cm.
Find its volume in terms of π.
26.
Find the slant height of the frustum in Q25.
27.
Find the curved surface area of the frustum in Q25.
28.
A sector of a circle of radius 14 cm and angle 120° is folded into a cone.
Find the radius of the cone.
29.
Find the height of the cone formed in Q28.
30.
Find the volume of the cone formed in Q28 in terms of π.
1.
Perimeter = 8 + 12 + 8 + (½ circumference of circle with d=12).
= 28 + ½(π×12) = 28 + 6π cm.
2.
Area = rectangle + semicircle = 12×8 + ½π×6².
= 96 + 18π cm².
3.
Area = square + semicircle = 10² + ½π×5².
= 100 + 12.5π cm².
4.
Area = rectangle + semicircle = 20×15 + ½π×7.5².
= 300 + 28.125π m².
5.
Perimeter semicircle = πr + 2r = 7π + 14 cm.
6.
Area quadrant = ¼πr² = ¼π×14² = 49π cm².
7.
Perimeter = sum of 4 semicircles = 2 circumferences (since opposite sides equal).
= 2π(16/2) + 2π(10/2) = 2π×8 + 2π×5 = 16π + 10π = 26π cm.
8.
Area = square + 4 semicircles = 14² + 2π×7² = 196 + 98π cm².
9.
Area of path = πR² − πr² = π(24² − 21²) = π(576 − 441) = 135π m².
10.
Area = π(R² − r²) = π(100 − 36) = 64π cm².
11.
Volume = πh(R² − r²) = π×15(100 − 49) = π×15×51 = 765π cm³.
12.
CSA = 2πh(R+r) = 2π×15(10+7) = 2π×15×17 = 510π cm².
13.
Big cone V = ⅓πr²h = ⅓π×7²×24 = 392π.
Small cone (h=8, r=7×8/24=7/3).
Vsmall = ⅓π(7/3)²×8 = 136/9 π.
Frustum V = 392π − 136/9π = (3528−136)/9 π = 3392/9 π cm³.
14.
Slant big = √(24²+7²)=√(576+49)=√625=25.
Slant small = √(8²+(7/3)²) = √(64+49/9)=√(625/9)=25/3.
Slant frustum = 25−25/3 = 50/3 cm.
15.
Height=30, base r=14.
Half height=15 → small cone radius=7.
Vsmall=⅓π7²×15=245π cm³.
16.
Vbig=⅓π14²×30=1960π.
Frustum=1960π−245π=1715π cm³.
17.
Cylinder V=πr²h=π×7²×10=490π.
Hemisphere V=2/3πr³=2/3π×343=686/3π.
Total= (490+686/3)π= (1470+686)/3 π=2156/3 π cm³.
18.
CSA= CSA cylinder+CSA hemisphere =2πrh+2πr².
=2π×7×10+2π×49=140π+98π=238π cm².
19.
Height=12+9=21 cm.
Volume=Vcone+Vhemi =⅓π9²×12+2/3π9³=324π+486π=810π cm³.
20.
Cylinder V=π7²×30=1470π.
Cone V=⅓π7²×24=392π.
Total=1862π cm³.
21.
Cuboid V=8×8×10=640.
Hemisphere V=2/3π4³=128/3π.
Total=640+128/3π cm³.
22.
Cuboid V=6×6×8=288.
Cylinder hole=πr²h=π×2²×8=32π.
Remaining=288−32π cm³.
23.
Cube V=8³=512.
Hole V=πr²h=π×9×8=72π.
Remaining=512−72π cm³.
24.
Sphere V=4/3π6³=288π.
Each hemi CSA=2πr².
Two hemispheres TSA=2×2π×36=144π.
Plus circular bases 2×πr²=72π.
Total SA=216π cm².
25.
Vfrustum=⅓πh(R²+r²+Rr)=⅓π×20(144+64+96)=⅓π×20×304= 6080/3 π cm³.
26.
Slant=√((R−r)²+h²)=√((12−8)²+20²)=√(16+400)=√416≈20.4 cm.
27.
CSA frustum=π(R+r)l=π(12+8)×20.4=20π×20.4=408π cm² approx.
28.
Arc length=120/360×2π×14=28/3π.
Circumference of base cone=2πr=28/3π → r=14/3 cm.
29.
Slant length=14 (radius of sector).
l²=h²+r² → h²=14²−(14/3)²=196−196/9= (1764−196)/9=1568/9.
h=√(1568)/3 ≈41/3 cm ≈13.7 cm.
30.
Volume=⅓πr²h=⅓π(14/3)²×(41/3)= (196/9×41/3)/3 π= 196×41/81 π≈ 7996/81 π ≈98.7π cm³.