Mathematics · Lesson 17

Mensuration I: plane shapes, arcs, sectors & chords

JAMBWAECNECO

What the syllabus asks

JAMB: Mensuration: lengths and areas of plane geometrical figures, length of arcs and chords, perimeters and areas of sectors and segments

WAEC: Mensuration: lengths and perimeters of plane figures, areas of triangles, quadrilaterals, circles and composite figures; arcs, sectors, chords and segments of circles

NECO: Mensuration: lengths and perimeters of plane figures, areas of triangles, quadrilaterals, circles and composite figures; arcs, sectors, chords and segments of circles

What you will learn

Find perimeters and areas of triangles, quadrilaterals and circles.
Use ½ab sin C and Heron’s formula for triangles.
Calculate arc lengths and perimeters of sectors.
Calculate sector areas and find angles or radii from them.
Find chord lengths and distances from the centre.
Find areas of segments and composite figures.

Key points

Perimeter in units; area in square units; 1 m² = 10,000 cm².
Triangle: ½bh, ½ab sin C, or Heron √[s(s−a)(s−b)(s−c)].
Parallelogram bh; trapezium ½(a + b)h; rhombus/kite ½d₁d₂.
Circle: C = 2πr, A = πr²; annulus π(R² − r²).
Arc = (θ/360)2πr; sector = (θ/360)πr²; sector perimeter = 2r + arc.
Perpendicular from centre bisects a chord; chord = 2r sin(θ/2).
Segment = sector − ½r² sin θ.
Similar figures: area ratio = (length ratio)²; a sector folds into a cone with arc = base circumference.

Start here: Perimeter and area: the basic ideas

Perimeter is the total distance round the edge of a shape. If a farmer in Benue wants to fence a plot, he needs its perimeter. Area measures how much surface the shape covers. If he wants to know how much land he can plant, he needs its area.

Perimeter is a length, so it is measured in cm, m or km. Area is measured in square units: cm², m², hectares (1 hectare = 10,000 m²).

Rectangle: perimeter = 2(l + b), area = l × b.
Square: perimeter = 4s, area = s².

Converting area units needs care. Since 1 m = 100 cm, 1 m² = 100 × 100 = 10,000 cm². Squaring the conversion factor is the key.

Worked example 17.1

A rectangular classroom floor is 9 m by 7 m. Find (a) its perimeter, (b) its area, (c) the cost of tiling it at ₦8,500 per m².

Perimeter = 2(9 + 7) = 32 m.
Area = 9 × 7 = 63 m².
Cost = 63 × 8,500 = ₦535,500.

32 m; 63 m²; ₦535,500

Worked example 17.2

A square has an area of 196 cm². Find its perimeter.

Side = √196 = 14 cm. We take the square root because area = side².
Perimeter = 4 × 14 = 56 cm.

56 cm

⚠️ Common mistake: converting 3 m² to 300 cm². It is 30,000 cm².

Try it yourself. A rectangular plot is 40 m long and has perimeter 130 m. Find its width and area.

Show solution

2(40 + b) = 130 → b = 25 m; area = 1,000 m².

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Practice questions

1. Find the area of a triangle with sides 5, 12 and 13 cm.

Show worked answer

Right-angled (5² + 12² = 13²): 30 cm².

2. Find the circumference of a circle of area 154 cm². (π = 22/7)

Show worked answer

r = 7: 44 cm.

3. Find the arc length for θ = 135°, r = 28 cm. (π = 22/7)

Show worked answer

135/360 × 176 = 66 cm.

4. A sector of radius 6 cm has angle 150°. Find its area. (π = 3.142)

Show worked answer

150/360 × 3.142 × 36 = 47.13 cm².

5. A chord of 10 cm is in a circle of radius 13 cm. Find its distance from the centre.

Show worked answer

√(169 − 25) = 12 cm.

Key words

Perimeter: The total length of the boundary of a figure.
Area: The amount of surface enclosed, in square units.
Radius: The distance from the centre of a circle to its edge.
Diameter: A chord through the centre; twice the radius.
Circumference: The perimeter of a circle.
Arc: A part of the circumference.
Minor arc: The shorter arc between two points.
Sector: The region between two radii and an arc.
Chord: A straight line joining two points on a circle.
Segment: The region between a chord and an arc.
Annulus: The ring between two concentric circles.
Subtend: A chord or arc subtends an angle at a point when its ends are joined to that point.

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