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Lesson 10 of 10

Measuring Space: Perimeter and Area · Lesson 10 of 10

Chapter Summary and Practice

“Add up the boundaries, areas, arcs and slices—then measure how much you remember.”

Learning Objectives

• Recall all major perimeter and area formulas. • Choose suitable methods for circle, triangle and quadrilateral problems. • Use Heron's and Brahmagupta's formulas correctly. • Connect arc length and sector area with central angle. • Solve mixed measurement problems with clear units.

This chapter connects one-dimensional measurement of boundaries with two-dimensional measurement of regions. It begins with perimeter and circumference, develops π and arc length, then builds area formulas for rectangles, parallelograms, triangles, cyclic quadrilaterals and circles.

Key Formulas and Results

IdeaFormula
Circle circumferenceC=2πr=πd
Arc lengthℓ=2πr·θ/360
Rectangle areaA=ab
Parallelogram areaA=bh
Triangle areaA=1/2 bh
Heron's formulaA=√[s(s−a)(s−b)(s−c)]
Circle areaA=πr²
Sector areaA=πr²·θ/360
Brahmagupta's formulaLaTeX

How to Choose the Right Method

Given informationBest method
Circle radius/diameter, boundary neededCircumference
Central angle and radius, curved length neededArc length
Base and perpendicular heightbh or 1/2bh
Three triangle sidesHeron's formula
Four sides of a cyclic quadrilateralBrahmagupta's formula
Circle radius, region neededπr²
Central angle and radius, sector regionSector area
Segment regionSector area minus triangle area
Common Mistakes

• Mixing perimeter units and area units, • Using diameter in place of radius without converting, • Forgetting the θ/360 factor for arcs and sectors, • Using a sloping side instead of perpendicular height, • Applying Brahmagupta's formula to a quadrilateral that is not known to be cyclic, • Treating 22/7 as exactly equal to π, • Forgetting to include straight radii when finding a sector's perimeter,

Guided Practice

Guided Example 1: Circumference

Problem
A circle has perimeter 44 cm. Find its radius using π≈22/7.

  1. 1.44=2×22/7×r.
  2. 2.44=(44/7)r.
  3. 3.r=7 cm.
Guided Example 2: Heron's Formula

Problem
Find the area of a triangle with sides 7,24,25.

  1. 1.s=(7+24+25)/2=28.
  2. 2.A=√[28×21×4×3].
  3. 3.A=√7056=84 cm².
Guided Example 3: Sector

Problem
Find the area of a 90° sector of radius 10 cm.

  1. 1.A=π×10²×90/360.
  2. 2.A=25π cm².
Guided Example 4: Area Scaling

Problem
The sides of a triangle are all doubled. How does its area change?

  1. 1.All linear dimensions scale by 2.
  2. 2.Area scales by the square of the scale factor.
  3. 3.New area=2²=4 times the original area.
Guided Example 5: Cyclic Rectangle

Problem
Use Brahmagupta's formula for a rectangle 6 cm by 8 cm.

  1. 1.Sides are 6,8,6,8, so s=14.
  2. 2.A=√[(14−6)(14−8)(14−6)(14−8)].
  3. 3.A=√(8×6×8×6)=48 cm².
  4. 4.This agrees with 6×8=48.

Practice Problems

Practice Questions
  1. A circle has circumference 66 cm. Find its radius using 22/7.
  2. Find the length of a 60° arc in a circle of radius 14 cm.
  3. Find the perimeter of a 75° sector of radius 14 cm, including both radii.
  4. A car tyre has diameter 56 cm. How far does it travel in 250 revolutions?
  5. A parallelogram has base 15 cm and height 8 cm. Find its area.
  6. A triangle has base 18 cm and height 11 cm. Find its area.
  7. Use Heron's formula for a triangle with sides 13 cm, 14 cm and 15 cm.
  8. An isosceles triangle has perimeter 40 cm and equal sides 15 cm. Find its area.
  9. A right triangle has area 54 cm² and one leg 12 cm. Find its perimeter.
  10. A cyclic quadrilateral has sides 5,5,12,12. Find its area using Brahmagupta's formula.
  11. Find the area of a circle of radius 14 cm.
  12. Find the area swept by a 10 cm clock hand in 15 minutes.
  13. A radius-10 cm circle has a chord subtending 90° at the centre. Find the minor sector area.
  14. A radius-15 cm circle has a chord subtending 60°. Find the minor segment area.
  15. Explain why doubling all lengths in a figure multiplies its area by 4.

Quiz

Quick check

Which expression gives arc length for central angle θ?

Quick check

Which formula can find a triangle's area using only its three side lengths?

Quick check

What extra condition is required before using Brahmagupta's four-side formula?

Quick check

If a circle's radius is tripled, its area becomes:

Quick check

Which statement about π is correct?

Before You Finish the Chapter

Check that you can distinguish perimeter from area, use circumference and arc length, explain what π represents, identify the correct height for parallelograms and triangles, apply Heron's and Brahmagupta's formulas, derive circle area conceptually, and calculate sectors and segments.

Key Takeaways

Key Takeaways

• Perimeter measures boundary; area measures region. • π links circumference and diameter and is irrational. • Arc length and sector area are controlled by central-angle fractions. • Triangle and parallelogram areas depend on perpendicular height. • Heron's and Brahmagupta's formulas use side lengths and semi-perimeter. • Circle area is πr². • Special cases and generalisation connect many formulas across mathematics.

Coming Next

This completes Measuring Space: Perimeter and Area. Continue by practising how to identify which measurement—length, perimeter or area—the problem is actually asking for.

Previous · Lesson 9

Area of Sectors and Segments

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End of chapter