Classwork Series and Exercises {Mathematics – SS1}: REVIEW OF SURFACE AREAS AND VOLUMES OF SOLID SHAPES

SSS 2 THIRD TERM MATHEMATICS WEEK TWO

Topic: REVIEW OF SURFACE AREAS AND VOLUMES OF SOLID SHAPES

Geometric Shape: Capsule

Volume = πr²((4/3)r + a), Surface Area

= 2πr(2r + a), Circumference = 2πr

Circular Cone

Volume = (1/3)πr²h, Slant Height = √(r² + h²), Lateral Surface Area = πrs = πr√(r² + h²), Base Surface Area = πr²

Total Surface Area
= L + B = πrs + πr² = πr(s + r) = πr(r + √(r² + h²))

Circular Cylinder

Volume = πr²h, Lateral Surface Area = 2πrh, Top Surface Area = πr², Bottom Surface Area = πr²

Total Surface Area
= L + T + B = 2πrh + 2(πr²) = 2πr(h+r)

Conical Frustum

Volume = (1/3)πh (r₁² + r₂² + (r₁ * r₂)), Slant Height = √((r₁ – r₂)² + h²),

Lateral Surface Area
= π(r₁ + r₂)s = π(r₁ + r₂)√((r₁ – r₂)² + h²)

Top Surface Area = πr₁², Base Surface Area = πr₂²

Total Surface Area
= π(r₁² + r₂² + (r₁ * r₂) * s)
= π[fusion_builder_container hundred_percent=”yes” overflow=”visible”][fusion_builder_row][fusion_builder_column type=”1_1″ background_position=”left top” background_color=”” border_size=”” border_color=”” border_style=”solid” spacing=”yes” background_image=”” background_repeat=”no-repeat” padding=”” margin_top=”0px” margin_bottom=”0px” class=”” id=”” animation_type=”” animation_speed=”0.3″ animation_direction=”left” hide_on_mobile=”no” center_content=”no” min_height=”none”][ r₁² + r₂² + (r₁ * r₂) * √((r₁ – r₂)² + h²) ]

Cube

Volume = a³, Surface Area = 6a², Face Diagonal (f) = a√2, Diagonal (d) = a√3

Hemisphere

Volume = (2/3)πr³, Circumference = 2πr, Curved Surface Area = 2πr², Base Surface Area = πr²

Total Surface Area= (2πr²) + (πr²) = 3πr²

Pyramid-Square

Volume = (1/3)a²h, Slant Height (s) = √(h² + (1/4)a²), Lateral Surface Area = a√(a² + 4h²), Base Surface Area = a²

Total Surface Area
= L + B = a² + a√(a² + 4h²))
= a(a + √(a² + 4h²))

Rectangular Prism

Volume = lwh, Surface Area = 2(lw + lh + wh), Diagonal (d) = √(l² + w² + h²)

Sphere

Volume = (4/3)πr³, Circumference = 2πr, Surface Area = 4πr²

Relationship between Sector and Cone

If we were to cut the cone up one side along the red line and roll it out flat, it would look something like the shaded pie-shaped section below.

1. This shaded section is actually part of a larger circle that has a radius of s, the slant height of the cone. (To flatten it, the cone was cut along the red lines, the length of this cut is the slant height of the cone.)

The area of the larger circle is therefore the area of a circle radius s, or πs²

2. The circumference of the larger circle, radius s is 2πs

3. The arc AB originally wrapped around the base of the cone, and so its length is the circumference of the base. Recall that circumference of a circle is given by 2πr Where r is the radius of the base of the cone.

4. The ratio of area x of the shaded sector to the area of the whole circle, is the same as the ratio of the arc AB to circumference of the whole circle.

Put as an equation x/area of large circle = AB/circumference of large circle. Substituting the values from above: x/πs² = 2πr/2πs. Canceling the 2π on the right and solving for x we get x = πrs

1. Finally, adding the areas of the base and the top part produces the final formula:  area = πrs + πr²

Solving Problems on Surface areas of Solid Shapes and Volumes

Examples

 

(a) Calculate the volume of the cuboid shown.

Volume = 4 × 18 × 5 = 360 m³

(b) Calculate the surface area of the cuboid shown.

Surface area = (2 x 4 x 18) + (2 x 4 x 5) + (2 x 5 x 18)

                         = 144 + 40 + 180

                         = 364 m² 

Calculate the volume and total surface area of the cylinder shown.

Volume = πr²h = π x 4² x 6 =96π

              = 301.5928947 cm³

              = 30 cm³ (to 3 significant figures)

Area of curved surface = 2πrh = 2 x π x 4 x 6

                                      = 48π

                                      = 150.7964474 cm²

Area of each end = πr² = π x 4²

                            = 16π

                               = 50.26548246 cm²

Total surface area = 150.7964474 + (2 x 50.26548236)

                                    = 251.3274123 cm²

                                = 251 cm² (to 3 significant figures)

Note: From the working we can see that the area of the curved surface is 48π, and that the area of each end is 16π. The total surface area is therefore

48π + (2 x 16π) = 80π = 251.3274123 cm²

                          = 251 cm² (to 3 significant figures)

Calculate the volume of this prism.

Area of end of prism = 12 x 8 x 6

                                  = 24 cm²

 Volume of prism = 24 x 10

                                  = 240 cm²

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