Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Mensuration
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Mensuration
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Subject: General Mathematics
Class: Senior Secondary 1
Term: 1st Term
Week: 9
Theme: Geometry
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Revise the components of a circle Find the length of arcs practically Find the length of arcs using for mula. Determine the perimeter of sectors of circles Determine the perimeter of segments of circles. Find the area of a sector Determine the area of a segment Determine the relationship between the sector of a circle and the surface area of a cone. Find the surface are as and volume of cube, cuboids, cylinder, cone, prisms, pyramids. Find surface area and volumes of fraction of a cone and pyramid. Find surface area and volume of compound shapes.
Compound shapes are formed by combining two or more basic geometric shapes. To find their surface area or volume, one must: Decompose the shape into simpler, recognizable components (e.g., a cylinder with a hemispherical top, a cuboid with a pyramid on top). Calculate the surface area/volume of each component. Add/Subtract the relevant areas/volumes, paying careful attention to shared surfaces (which are not included in the total surface area).
Example 8 (Compound Shape Volume): A storage unit is made of a cylindrical base with a hemispherical roof. The cylinder has a radius of 3 m and a height of 5 m. Calculate the total volume of the storage unit. (Use π = 3.142)
Solution: Volume of Cylinder: V_cylinder = πr2h = 3.142 (3)2 5 V_cylinder = 3.142 9 5 = 141.39 m3 Volume of Hemisphere: A hemisphere is half of a sphere. Volume of Sphere = (4/3)πr
3. V_hemisphere = (1/2) * (4/3)πr3 = (2/3)πr3 V_hemisphere = (2/3) 3.142 (3)3 V_hemisphere = (2/3) 3.142 27 V_hemisphere = 2 3.142 9 = 56.556 m3 Total Volume: Total Volume = V_cylinder + V_hemisphere Total Volume = 141.39 + 56.556 = 197.946 m3 The total volume of the storage unit is approximately 197.95 m
3. A circle is a set of all points in a plane that are equidistant from a central point.
Centre (O): The fixed point from which all points on the circle are equidistant.
Radius (r): A line segment connecting the centre to any point on the circle. All radii of a circle are equal.
Diameter (d): A line segment passing through the centre and connecting two points on the circle. `d = 2r`.
Circumference (C): The distance around the circle. `C = 2πr = πd`.
Chord: A line segment connecting any two points on the circle. The diameter is the longest chord.
Arc: A portion of the circumference of a circle.
Minor Arc: An arc shorter than half the circumference.
Major Arc: An arc longer than half the circumference.
Sector: A region of a circle bounded by two radii and the arc connecting their endpoints.
Minor Sector: Bounded by two radii and a minor arc.
Major Sector: Bounded by two radii and a major arc.
Segment: A region of a circle bounded by a chord and the arc subtended by the chord.
Minor Segment: Bounded by a chord and a minor arc.
Major Segment: Bounded by a chord and a major arc. The length of an arc is a fraction of the circle's total circumference, determined by the angle the arc subtends at the centre. 2.2.
1. Practical Method: For a given sector, students can measure the radius (r) and the angle (θ) at the centre using a protractor. The arc can be measured by carefully placing a string along its curve and then measuring the string's length with a ruler. This provides an empirical understanding before introducing the formula. 2.2.
2. Using Formula: If the angle subtended by the arc at the centre is θ (in degrees) and the radius of the circle is r, the length of the arc (L) is given by: `L = (θ / 360°) * 2πr` Where `π` (pi) is a mathematical constant approximately equal to `22/7` or `3.142`.
Example 1: A circular field has a radius of 14 meters. A section of the fence forms an arc that subtends an angle of 90° at the centre. Find the length of this arc. (Use π = 22/7)
Solution: Given: r = 14 m, θ = 90° L = (θ / 360°) * 2πr L = (90 / 360) 2 (22/7) * 14 L = (1/4) 2 22 * 2 L = (1/4) * 88 L = 22 meters The length of the arc is 22 meters. The perimeter of a sector is the sum of the lengths of the two radii and the arc length. Perimeter of Sector = Arc Length + 2 * radius `P = (θ / 360°) * 2πr + 2r` Example 2: A sector of a circular mat has a radius of 7 cm and an angle of 120° at the centre. Determine its perimeter. (Use π = 22/7)
Solution: Given: r = 7 cm, θ = 120° First, find the arc length (L): L = (θ / 360°) * 2πr L = (120 / 360) 2 (22/7) * 7 L = (1/3) 2 22 L = 44/3 cm ≈ 14.67 cm Now, find the perimeter (P) of the sector: P = L + 2r P = 44/3 + 2 * 7 P = 44/3 + 14 P = (44 + 42) / 3 P = 86/3 cm ≈ 28.67 cm The perimeter of the sector is approximately 28.67 cm.
Building and Construction (Housing/Infrastructure): Roofing: Architects and builders use surface area calculations to determine the number of roofing sheets (e.g., trapezoidal sheets for a frustum-shaped dome or triangular sheets for a pyramidal roof) needed for buildings, churches, or mosques. Volume calculations are essential for estimating concrete and cement needed for foundations, pillars, and slabs (cuboids, cylinders).
Water Storage: Calculating the volume of cylindrical overhead tanks or cuboidal underground reservoirs helps determine water capacity for households, schools, and communities.
Road Construction: Estimating the volume of asphalt or laterite needed for road construction involves calculations of volumes of prisms or irregular shapes approximated by simpler forms.
Agriculture and Rural Development: Silo Capacity: Farmers can determine the storage capacity (volume) of cylindrical silos for grains (maize, rice) or other produce, ensuring efficient harvesting and storage planning.
Irrigation: Calculating the flow rate and volume of water in cylindrical pipes or trapezoidal canals for irrigation systems to optimize water distribution to farmlands.
Crop Yield Estimation: Mensuration principles can be used to estimate crop yields per unit area, e.g., in circular or rectangular farm plots, aiding in agricultural planning and resource management.
Manufacturing and Packaging: Product Design: Companies design product packaging (e.g., cylindrical Milo tins, cuboidal Indomie cartons, conical pap-wraps) using mensuration to optimize material usage (surface area) and product quantity (volume). This impacts production cost and shelf space.
Crafts and Artistry: Artisans creating pottery (frustums, cylinders), calabash carvings (hemispheres), or traditional fabrics with circular motifs (arc lengths, sector areas) implicitly apply mensuration principles in their designs and material estimation.