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Chapter 5: Problem 204

Find the exact length (in terms of \(\pi)\) (i) of a semicircle of radius \(r\); (ii) of a quarter circle of radius \(r\); (iii) of the length of an arc of a circle of radius \(r\) that subtends an angle \(\theta\) radians at the centre. \(\wedge\) In the next problem we follow a similar sequence of steps to conclude that the quotient \(L=\frac{\text { area of circle of radius } r}{r^{2}}\) is also constant. The surprise lies in the fact that this different constant is so closely related to the previous constant \(K\).

Short Answer

Expert verified
(i) \(\pi r\); (ii) \(\frac{\pi r}{2}\); (iii) \(r\theta\); (iv) \(L = \pi\).

Step by step solution

01

Calculate Semi-Circle Perimeter

The perimeter of a semicircle consists of the half circumference of the circle plus the diameter.The full circumference of a circle is calculated as \(2\pi r\). So, the perimeter of a semicircle is given by \(\pi r + 2r\). Since we are asked for the exact length in terms of \(\pi\), the perimeter is \(\pi r\) (the curved part only) without considering the diameter.
02

Calculate Quarter-Circle Perimeter

The perimeter of a quarter of a circle consists of a quarter of the circle's circumference and the radius lines on both sides. The formula for the full circumference is \(2\pi r\). For a quarter-circle, the arc length is \(\frac{1}{4}\cdot 2\pi r = \frac{\pi r}{2}\). Thus, the exact length for the arc part of the perimeter is \(\frac{\pi r}{2}\) without considering the radii.
03

Calculate Arc Length for Angle \(\theta\)

The arc length of a circle section that subtends an angle \(\theta\) radians is a fraction of the circle's circumference. The full circle circumference is \(2\pi r\). The arc length is given by \(r\cdot \theta\), where \(\theta\) is in radians.
04

Calculate Constant L for Circle Area

The area of a circle is \(\pi r^2\). We need to find the quotient \(L = \frac{\text{area of circle of radius } r}{r^2}\), which would be \(L = \frac{\pi r^2}{r^2} = \pi\). This shows \(L\) is also constant and equals \(\pi\), closely related to the constant \(K\) from the perimeters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circle Circumference
The circumference of a circle is the total distance around it. Think of it as the circle's perimeter. To calculate the circumference, we use the formula:
  • Full Circumference: \(2\pi r\)
Here, \(r\) represents the radius of the circle, and \(\pi\) approximately equals 3.14159.
This formula shows that the circumference is directly proportional to the radius. The larger the radius, the longer the circumference.
In the case of a semicircle or a quarter circle, only a portion of the full circumference is considered for the arc. This is why we used fractions in the calculations:
  • Semi-circle: \(\pi r\)
  • Quarter circle: \(\frac{\pi r}{2}\)
Arc Length
Arc length is the distance along a curved line making up part of the circumference of a circle. It's useful when dealing with segments of a circle rather than the entire shape.
Why calculate arc length? Often, only part of a circle is needed for real-world applications, like wheels or roundabouts.
The formula for calculating the arc length when you have the central angle \(\theta\) in radians is:
  • Arc Length: \(r \cdot \theta\)
This simple formula relies on the fact that the angle \(\theta\) is measured in radians. Remember that the arc length is part of the circle's circumference, which ties back to the full circle formula \(2\pi r\).
Circle Area
The area of a circle measures the space enclosed within its boundary. It's important in scenarios like calculating materials needed to cover a circular region.
The circle area formula is:
  • Area: \(\pi r^2\)
This formula shows the area is proportional to the square of the radius. A bigger circle means a much larger area due to this square relationship.
In our problem, we found that dividing the area of a circle by the square of its radius gives the constant \(\pi\). This result is a key feature of circles, revealing an intrinsic geometry property.
Radius
The radius of a circle is the distance from its center to any point on its edge. It's a pivotal parameter in circles, helping determine other properties.
A longer radius means a larger circle. Since many formulas for circles include the radius, understanding it is crucial.
  • Circumference involves \(r\)
  • Area uses \(r^2\)
  • Arc length relies on \(r\)
In this exercise, all calculations regarding the semicircle, quarter-circle, and arc length are based on knowing the radius. The radius therefore becomes the master key to unlocking other circle measurements.
Radians
Radians provide a natural way of measuring angles, especially in calculations involving circles. Unlike degrees, radians stem directly from the circle's properties.
In a full circle, there are \(2\pi\) radians, equivalent to 360 degrees. Therefore, one radian equals \(\frac{180}{\pi}\) degrees.
Radians simplify the computation of arc lengths. Instead of converting degrees to part of the circumference, you can calculate using:
  • Arc Length: \(r \cdot \theta\)
where \(\theta\) is in radians. This makes radians the go-to unit in trigonometry and circle-related problems.

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Most popular questions from this chapter

Given a spherical triangle \(\triangle A B C\) on the unit sphere with centre \(O,\) such that \(\angle B A C\) is a right angle, and such that \(\underline{A B}\) has length \(c\), and \(A C\) has length \(b\). (a) We have (rightly) referred to \(b\) and \(c\) as 'lengths'. But what are they really? (b) We want to know how the inputs \(b\) and \(c\) determine the value of the length \(a\) of the arc \(\underline{B C} ;\) that is, we are looking for a function with inputs \(b\) and \(c\), which will allow us to determine the value of the "output" \(a\). Think about the answer to part (a). What kind of standard functions do we already know that could have inputs \(b\) and \(c ?\) (c) Suppose \(c=0 \neq b\). What should the output \(a\) be equal to? (Similarly if \(b=0 \neq c .)\) Which standard function of \(b\) and of \(c\) does this suggest is involved? (d)(i) Suppose \(\angle B=\angle C=\frac{\pi}{2},\) what should the output \(a\) be equal to? (ii) Suppose \(\angle B=\frac{\pi}{2},\) but \(\angle C\) (and hence \(c\) ) is unconstrained. The output \(a\) is then determined \(-\) but the formula must give this fixed output for different values of \(c\). What does this suggest as the "simplest possible" formula for \(a ?\)

You are given a pyramid \(A B C D\) with all three faces meeting at \(A\) being right angled triangles with right angles at \(A .\) Suppose \(\underline{A B}=b,\) \(\underline{A C}=c, \underline{A D}=d\) (a) Calculate the areas of \(\triangle A B C, \triangle A C D, \triangle A D B\) in terms of \(b, c, d\). (b) Calculate the area of \(\triangle B C D\) in terms of \(b, c, d\). (c) Compare your answer in part (b) with the sum of the squares of the three areas you found in part (a).

A pyramid \(A B C D E,\) with apex \(A\) and square base \(B C D E\) of side length \(b\), is cut parallel to the base at height \(d\) above the base, leaving a frustum of a pyramid, with square upper face of side length \(a\). Find a formula for the volume of the resulting solid (in terms of \(a, b,\) and \(d) . \quad \triangle\)

The point \(P\) lies inside a circle. Two secants from \(P\) meet the circle at \(A, B\) and at \(C, D\) respectively. Prove that $$\underline{P A} \times \underline{P B}=\underline{P C} \times \underline{P D}$$ We end our summary of the foundations of Euclidean geometry by deriving the familiar formula for the area of a trapezium and its 3-dimensional analogue, and a formulation of the similarity criteria which is often attributed to Thales (Greek \(6^{\text {th }}\) century BC).

(a) Let \(A B C D\) be a regular tetrahedron with edges of length 2. Calculate the (exact) angle between the two faces \(A B C\) and \(D B C\). (b) We know that in \(2 \mathrm{D}\) five equilateral triangles fit together at a point leaving just enough of an angle to allow a sixth triangle to fit. How many identical regular tetrahedra can one fit together, without overlaps around an edge, so that they all share the edge \(\underline{B C}\) (say)?
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