Mathematics: Equations, Perimeter, Area, and Circle Properties

Comprehensive practice flashcards covering circle vocabulary, circumference, area of circles, semicircles and compound shapes, perimeter problems, curriculum content indices, and rearranging multi-step algebraic formulae.

According to the Equations unit contents, on which page is "Solving One Step Equations" reviewed?

Page 2

Which topic in the Equations unit is designated as an "extra challenge" on Page 4?

Square Mystery

On which page does the Equations unit cover "Forming & Solving Equations (Angles)"?

Page 9

Which page of the Equations unit introduces the visual method of "Bar Models"?

Page 22

What two types of double-sided equations are distinguished in the unit contents on Pages 21 and 23?

Double Sided Equations (Positive Coefficients) on Page 21 and Double Sided Equations (Negative Coefficients) on Page 23

In Exercise 8g, Question 1a, how is the perimeter equation set up for a rectangle with adjacent sides $3x - 1$ and $2x$ and a perimeter of $32$?

$$2(3x - 1) + 2(2x) = 32$$

In Exercise 8g, Question 1c, a triangle has side lengths $8 - m$, $9 - m$, and $10 - m$, and a perimeter of $12$. What is the value of $m$?

$m = 5$, because $(8 - m) + (9 - m) + (10 - m) = 27 - 3m = 12$, giving $3m = 15$

In Exercise 8g, Question 2, a gate has a width of $2.4\,\text{m}$ and a height of $1\,\text{m}$. What is the length of its diagonal rod?

$2.6\,\text{m}$, calculated using Pythagoras' theorem: $\sqrt{(2.4)^2 + 1^2} = \sqrt{5.76 + 1} = \sqrt{6.76} = 2.6\,\text{m}$

In Exercise 8g, Question 3a, a right-angled triangle with a height of $5\,\text{cm}$ and a base of $2x\,\text{cm}$ has an area of $30\,\text{cm}^2$. What is the value of $x$?

$x = 6$, because $\frac{1}{2} \times 2x \times 5 = 5x = 30$

In Exercise 8g, Question 3b, what is the perimeter of a right-angled triangle with perpendicular legs measuring $5\,\text{cm}$ and $12\,\text{cm}$?

$30\,\text{cm}$, calculated as $5 + 12 + \sqrt{5^2 + 12^2} = 5 + 12 + 13 = 30\,\text{cm}$

In Exercise 8g, Question 4, what does the term "regular" mean when referring to regular hexagons?

All sides are equal

In Exercise 8g, Question 5, two squares each with an area of $36\,\text{cm}^2$ are placed side-by-side to form a rectangle. What is the perimeter of the rectangle?

$36\,\text{cm}$, because each square has side length $\sqrt{36} = 6\,\text{cm}$, giving rectangle dimensions of $12\,\text{cm} \times 6\,\text{cm}$ and a perimeter of $2(12 + 6) = 36\,\text{cm}$

In Exercise 8g, Question 6, the area of a parallelogram with base $8$ and height $h$ is four times the area of a triangle with base $6$ and height $3$. What is the value of $h$?

$h = 4.5$, because the triangle area is $\frac{1}{2} \times 6 \times 3 = 9$, four times this is $36$, and $8h = 36$ gives $h = 4.5$

In Page 39, Question 7, eight identical small rectangles are stacked to form a larger rectangle with a total height of $16\,\text{cm}$. If four narrow sides form this height, what is the shorter side of each rectangle?

$4\,\text{cm}$, calculated as $16 \div 4 = 4\,\text{cm}$

In Page 39, Question 8, what are the dimensions of the hockey pitch that Laura needs to paint?

A rectangle measuring $91.4\,\text{m}$ by $55\,\text{m}$ with two semicircular areas of radius $29.2\,\text{m}$

In Page 39, Question 9, what is the length of the hypotenuse of an isosceles right-angled triangle with an area of $100\,\text{cm}^2$?

$20\,\text{cm}$, because $\frac{1}{2}a^2 = 100 \implies a^2 = 200$, so the hypotenuse is $\sqrt{a^2 + a^2} = \sqrt{400} = 20\,\text{cm}$

In Page 39, Question 10, two identical overlapping squares have an overlapping area of $14\,\text{cm}^2$ and a non-overlapping area of $70\,\text{cm}^2$. What is the width of each square?

$7\,\text{cm}$, because the area of each square is $\frac{70 + 2(14)}{2} = 49\,\text{cm}^2$, giving a side length of $\sqrt{49} = 7\,\text{cm}$

In Page 39, Question 11a, each vertex of a square with area $36\,\text{cm}^2$ lies on the circumference of a circle. What is the diameter of the circle correct to 3 significant figures?

$8.49\,\text{cm}$, which is the diagonal of the square: $\sqrt{6^2 + 6^2} = \sqrt{72} \approx 8.49\,\text{cm}$

In Page 39, Question 11b, what is the area of a circle circumscribing a square of area $36\,\text{cm}^2$, correct to 3 significant figures?

$56.5\,\text{cm}^2$, calculated as $\pi \times \left(\frac{\sqrt{72}}{2}\right)^2 = 18\pi \approx 56.5\,\text{cm}^2$

In Page 39, Question 12, what general formula is used to calculate the area of a trapezium?

$\text{Area} = \frac{a + b}{2} \times h$, where $a$ and $b$ are the lengths of the parallel sides and $h$ is the perpendicular height

In Page 39, Question 12a, which algebraic expression represents one of the parallel bases of the trapezium?

$$7x - 6$$

According to the Contents page for the Perimeter, Area, and 3D Shapes unit, on which page does "Area of Circles, Semi-Circles & Quarter Circles" begin?

Page 9

On which page does the unit cover "Names & Properties of 3d Shapes"?

Page 22

What topic is covered immediately before "Units of Volume and Capacity" (Page 30)?

Surface Area (Page 28)

On which page is "Volume of Cubes & Cuboids" introduced?

Page 32

Which topic is designated as an extra challenge on Page 34 of the 3D Shapes unit?

Mixed & Combined Surface Area & Volume of Cuboids

On which page does "Volume of Prisms & Cylinders" appear?

Page 36

What is the definition of circumference?

The curve all the way round the circle

What is the definition of a radius of a circle, and what is its plural?

A straight line from the centre to the circumference; plural: radii

What is the definition of a diameter of a circle?

A straight line all the way across a circle, passing through the centre

What is the definition of a chord?

A straight line from one point on the circumference to another

What is the definition of an arc of a circle?

A curved part of the circumference

What is the definition of a tangent to a circle?

A straight line outside the circle that meets the circumference at a point, but does not cross it

What is the definition of a sector of a circle?

A 2D shape formed from an arc of a circle and two radii

What is the definition of a segment of a circle?

A 2D shape formed from an arc of a circle and a chord

Which two straight lines form the straight boundaries of a circle's sector?

Two radii

Which straight line of a circle is always double the length of the radius?

The diameter

Among a chord, radius, tangent, arc, and diameter, which one is not a straight line?

Arc

What is the formula for the circumference of a circle in terms of its diameter?

$\text{Circumference} = \pi \times \text{diameter}$ ($C = \pi d$)

What is the formula for the circumference of a circle in terms of its radius?

$\text{Circumference} = 2 \times \pi \times \text{radius}$ ($C = 2\pi r$)

What constant value is approximated by $\pi$ (pi) in the "learn by heart" section?

$\pi$ is a constant equal to $3.14\dots$

In Exercise 4p, Question 1a, what is the circumference of a circle with a radius of $5\,\text{cm}$, correct to 1 decimal place?

$31.4\,\text{cm}$, calculated as $2 \times \pi \times 5 \approx 31.4\,\text{cm}$

In Exercise 4p, Question 1c, what is the circumference of a circle with a diameter of $8\,\text{m}$, correct to 1 decimal place?

$25.1\,\text{m}$, calculated as $\pi \times 8 \approx 25.1\,\text{m}$

In Page 28, Question 2, Jared does not have a calculator. Which calculation can he use to estimate the circumference of a circle with radius $5.9\,\text{cm}$?

Calculation B: $2 \times 3 \times 6$

In Page 28, Question 3, what is the circumference of a circle with a diameter of $8\,\text{cm}$, correct to 1 decimal place?

$25.1\,\text{cm}$, calculated as $\pi \times 8 \approx 25.1\,\text{cm}$

In Page 28, Question 4, what is the circumference of a circle with a radius of $2.4\,\text{cm}$, correct to 1 decimal place?

$15.1\,\text{cm}$, calculated as $2 \times \pi \times 2.4 \approx 15.1\,\text{cm}$

In Page 28, Question 5, a circle is inscribed in a square with a perimeter of $20\,\text{cm}$. What is the circumference of the circle correct to 1 decimal place?

$15.7\,\text{cm}$, because the square has side length $20 \div 4 = 5\,\text{cm}$, giving a diameter of $5\,\text{cm}$ and $C = \pi \times 5 \approx 15.7\,\text{cm}$

In Page 28, Question 6, a circle touches two sides of a rectangle that has an area of $24\,\text{cm}^2$ and a height of $6\,\text{cm}$. What is the circumference of the circle to 1 decimal place?

$12.6\,\text{cm}$, because the rectangle's width is $24 \div 6 = 4\,\text{cm}$, making the circle's diameter $4\,\text{cm}$, and $C = \pi \times 4 \approx 12.6\,\text{cm}$

In Page 28, Question 7, which is larger: the circumference of a circle with diameter $10\,\text{cm}$, or the total circumference of two circles each with diameter $5\,\text{cm}$?

Neither; they are equal, because $\pi \times 10 = 10\pi$ and $(\pi \times 5) + (\pi \times 5) = 10\pi$

In Mixed Shapes (Page 28), what is the perimeter of Shape A, an equilateral triangle with side length $7\,\text{cm}$?

$21\,\text{cm}$, calculated as $3 \times 7 = 21\,\text{cm}$

In Mixed Shapes (Page 28), what is the perimeter of Shape B, a rectangle measuring $5\,\text{cm}$ by $2\,\text{cm}$?

$14\,\text{cm}$, calculated as $2(5 + 2) = 14\,\text{cm}$

In Mixed Shapes (Page 28), what is the perimeter of Shape C, a regular octagon with side length $2\,\text{cm}$?

$16\,\text{cm}$, calculated as $8 \times 2 = 16\,\text{cm}$

In Mixed Shapes (Page 28), what is the perimeter of Shape D, a kite with side lengths $4\,\text{cm}$ and $6\,\text{cm}$?

$20\,\text{cm}$, calculated as $2(4 + 6) = 20\,\text{cm}$

What is the formula for the area of a circle?

$\text{Area} = \pi \times \text{radius}^2$ ($A = \pi r^2$)

According to the instruction box on Page 29, how do you find the radius when given the diameter?

To find the radius, half the diameter ($\text{Radius} = \text{diameter} \div 2$)

In the worked example on Page 29, what is the area of a circle with radius $5\,\text{cm}$, rounded to 3 significant figures?

$78.5\,\text{cm}^2$, calculated as $\pi \times 5^2 = 78.5398\dots \approx 78.5\,\text{cm}^2$

In the worked example on Page 29, what is the area of a circle with diameter $13\,\text{m}$, rounded to 3 significant figures?

$133\,\text{m}^2$, calculated with radius $13 \div 2 = 6.5\,\text{m}$ as $\pi \times 6.5^2 = 132.7322\dots \approx 133\,\text{m}^2$

In Exercise 4q, Question 1a, what is the area of a circle with a radius of $3\,\text{cm}$, correct to 1 decimal place?

$28.3\,\text{cm}^2$, calculated as $\pi \times 3^2 = 9\pi \approx 28.3\,\text{cm}^2$

In Exercise 4q, Question 1b, what is the area of a circle with a diameter of $12\,\text{cm}$, correct to 1 decimal place?

$113.1\,\text{cm}^2$, calculated with radius $6\,\text{cm}$ as $\pi \times 6^2 = 36\pi \approx 113.1\,\text{cm}^2$

In Exercise 4q, Question 2, a circle is inscribed within a square with a perimeter of $88\,\text{cm}$. What is the area of the circle correct to 1 decimal place?

$380.1\,\text{cm}^2$, because the square has side length $88 \div 4 = 22\,\text{cm}$, giving a radius of $11\,\text{cm}$ and $\text{Area} = \pi \times 11^2 \approx 380.1\,\text{cm}^2$

In Exercise 4q, Question 3a, what is the area of the smaller circle in a pair of concentric circles with radii $3\,\text{cm}$ and $5\,\text{cm}$, to 1 decimal place?

$28.3\,\text{cm}^2$, calculated as $\pi \times 3^2 \approx 28.3\,\text{cm}^2$

In Exercise 4q, Question 3c, what is the shaded ring area between two concentric circles with radii $3\,\text{cm}$ and $5\,\text{cm}$, correct to 1 decimal place?

$50.3\,\text{cm}^2$, calculated as $\pi \times 5^2 - \pi \times 3^2 = 25\pi - 9\pi = 16\pi \approx 50.3\,\text{cm}^2$

In the Page 31 worked example, how is the area of a semicircle with a straight boundary of $7\,\text{cm}$ calculated?

Radius = $7 \div 2 = 3.5\,\text{cm}$; Area of full circle = $\pi \times 3.5^2 = 38.484\dots\,\text{cm}^2$; Area of semicircle = $38.484 \div 2 \approx 19.2\,\text{cm}^2$

In the Page 31 worked example, how is the area of a quarter circle with radius $6\,\text{cm}$ calculated?

Area of full circle = $\pi \times 6^2 = 113.097\dots\,\text{cm}^2$; Area of quarter circle = $113.097 \div 4 \approx 28.3\,\text{cm}^2$

In Exercise 4r, Question 1a, what is the area of a semicircle with diameter $10\,\text{cm}$, correct to 1 decimal place?

$39.3\,\text{cm}^2$, calculated with radius $5\,\text{cm}$ as $\frac{\pi \times 5^2}{2} = 12.5\pi \approx 39.3\,\text{cm}^2$

In Exercise 4r, Question 1b, what is the area of a semicircle with diameter $6\,\text{cm}$, correct to 1 decimal place?

$14.1\,\text{cm}^2$, calculated with radius $3\,\text{cm}$ as $\frac{\pi \times 3^2}{2} = 4.5\pi \approx 14.1\,\text{cm}^2$

In Exercise 4r, Question 1c, what is the area of a quarter circle with radius $5\,\text{cm}$, correct to 1 decimal place?

$19.6\,\text{cm}^2$, calculated as $\frac{\pi \times 5^2}{4} = 6.25\pi \approx 19.6\,\text{cm}^2$

In Exercise 4r, Question 1d, what is the area of a quarter circle with radius $8\,\text{cm}$, correct to 1 decimal place?

$50.3\,\text{cm}^2$, calculated as $\frac{\pi \times 8^2}{4} = 16\pi \approx 50.3\,\text{cm}^2$

In Exercise 4r, Question 1e, what is the area of a semicircle with diameter $15\,\text{cm}$, correct to 1 decimal place?

$88.4\,\text{cm}^2$, calculated with radius $7.5\,\text{cm}$ as $\frac{\pi \times 7.5^2}{2} = 28.125\pi \approx 88.4\,\text{cm}^2$

In Exercise 4r, Question 2a, what is the area of a shaded quarter of a circle that has a diameter of $14\,\text{cm}$, correct to 1 decimal place?

$38.5\,\text{cm}^2$, calculated with radius $7\,\text{cm}$ as $\frac{\pi \times 7^2}{4} = 12.25\pi \approx 38.5\,\text{cm}^2$

In Exercise 4r, Question 3, a circular sector is formed within a square that has a perimeter of $36\,\text{cm}$. What is the area of this quarter-circle sector to 1 decimal place?

$63.6\,\text{cm}^2$, because the square has side length $36 \div 4 = 9\,\text{cm}$, giving an area of $\frac{\pi \times 9^2}{4} = 20.25\pi \approx 63.6\,\text{cm}^2$

In the Page 33 example, how is the shaded area calculated for a $7\,\text{cm} \times 3\,\text{cm}$ rectangle joined to a quarter circle of radius $3\,\text{cm}$?

Rectangle area = $7 \times 3 = 21\,\text{cm}^2$; Quarter circle area = $\frac{\pi \times 3^2}{4} \approx 7.06858\,\text{cm}^2$; Total shaded area = $21 + 7.06858 \approx 28.1\,\text{cm}^2$

In the Page 33 subtraction example, how is the shaded area calculated when a semicircle of diameter $10\,\text{cm}$ is cut from a $10\,\text{cm} \times 5\,\text{cm}$ rectangle?

Rectangle area = $10 \times 5 = 50\,\text{cm}^2$; Semicircle area = $\frac{\pi \times 5^2}{2} \approx 39.2699\,\text{cm}^2$; Shaded area = $50 - 39.2699 \approx 10.7\,\text{cm}^2$

In Exercise 4s, Question 1a, what is the total area of a composite shape made of a $6\,\text{cm} \times 6\,\text{cm}$ square attached to a semicircle of diameter $6\,\text{cm}$, to 1 decimal place?

$50.1\,\text{cm}^2$, calculated as $36 + \frac{\pi \times 3^2}{2} = 36 + 14.137\dots \approx 50.1\,\text{cm}^2$

In Exercise 4s, Question 1b, what is the area of a composite shape comprising a right triangle with base $9\,\text{cm}$ and height $6\,\text{cm}$ joined to a quarter circle of radius $6\,\text{cm}$?

$55.3\,\text{cm}^2$, calculated as $\left(\frac{1}{2} \times 9 \times 6\right) + \frac{\pi \times 6^2}{4} = 27 + 9\pi \approx 55.3\,\text{cm}^2$

In Exercise 4s, Question 1c, what is the total combined area of two identical quarter circles of radius $7\,\text{cm}$, correct to 1 decimal place?

$77.0\,\text{cm}^2$, which equals the area of a semicircle with radius $7\,\text{cm}$: $\frac{\pi \times 7^2}{2} = 24.5\pi \approx 77.0\,\text{cm}^2$

In Exercise 4s, Question 1d, which basic geometric figures make up the compound shape with an overall height of $12\,\text{cm}$?

A rectangle and a semicircle

In Exercise 4s, Question 1f, what is the total area of a composite shape formed by an $8\,\text{cm} \times 8\,\text{cm}$ square and a three-quarter circle with radius $8\,\text{cm}$, correct to 1 decimal place?

$214.8\,\text{cm}^2$, calculated as $8^2 + \left(\frac{3}{4} \times \pi \times 8^2\right) = 64 + 48\pi \approx 214.8\,\text{cm}^2$

In Exercise 4s, Question 2, a circle is inscribed in a square of side length $22\,\text{cm}$. What is the shaded area outside the circle but inside the square, correct to 1 decimal place?

$103.9\,\text{cm}^2$, calculated as $22^2 - (\pi \times 11^2) = 484 - 121\pi \approx 103.9\,\text{cm}^2$

In Page 34, Question 3a, a quarter circle of radius $4\,\text{cm}$ is removed from a $4\,\text{cm} \times 4\,\text{cm}$ square. What is the remaining shaded area to 1 decimal place?

$3.4\,\text{cm}^2$, calculated as $4^2 - \frac{\pi \times 4^2}{4} = 16 - 4\pi \approx 3.4\,\text{cm}^2$

In Page 34, Question 3b, a semicircle of diameter $6\,\text{cm}$ is removed from a rectangle measuring $3\,\text{cm} \times 6\,\text{cm}$. What is the remaining shaded area to 1 decimal place?

$3.9\,\text{cm}^2$, calculated as $(3 \times 6) - \frac{\pi \times 3^2}{2} = 18 - 4.5\pi \approx 3.9\,\text{cm}^2$

In Page 34, Question 3c, four corner quarter circles of radius $5\,\text{cm}$ are removed from a $10\,\text{cm} \times 10\,\text{cm}$ square. What is the remaining shaded area to 1 decimal place?

$21.5\,\text{cm}^2$, calculated as $10^2 - (\pi \times 5^2) = 100 - 25\pi \approx 21.5\,\text{cm}^2$

In Page 34, Question 3e, a right-angled triangle with legs measuring $8\,\text{cm}$ is cut from a quarter circle of radius $8\,\text{cm}$. What is the shaded area to 1 decimal place?

$18.3\,\text{cm}^2$, calculated as $\frac{\pi \times 8^2}{4} - \left(\frac{1}{2} \times 8 \times 8\right) = 16\pi - 32 \approx 18.3\,\text{cm}^2$

In Page 34, Question 4a, a rectangle of length $12\,\text{cm}$ encloses two identical tangent circles. What is the shaded area surrounding the circles to 1 decimal place?

$15.5\,\text{cm}^2$, because each circle has diameter $6\,\text{cm}$ (radius $3\,\text{cm}$) and rectangle height is $6\,\text{cm}$, giving $(12 \times 6) - 2(\pi \times 3^2) = 72 - 18\pi \approx 15.5\,\text{cm}^2$

In Page 34, Question 4b, a rectangle of height $10\,\text{cm}$ contains three identical tangent circles. What is the shaded area surrounding the circles to 1 decimal place?

$64.4\,\text{cm}^2$, because each circle has diameter $10\,\text{cm}$ (radius $5\,\text{cm}$) and rectangle length is $30\,\text{cm}$, giving $(30 \times 10) - 3(\pi \times 5^2) = 300 - 75\pi \approx 64.4\,\text{cm}^2$

In Page 34, Question 5a, what is the area of the smaller circle in a pair of concentric circles with an inner radius of $3\,\text{m}$ and a ring width of $2\,\text{m}$, to 1 decimal place?

$28.3\,\text{m}^2$, calculated as $\pi \times 3^2 \approx 28.3\,\text{m}^2$

In Page 34, Question 5b and 5c, what are the area of the larger circle and the shaded ring area for concentric circles with inner radius $3\,\text{m}$ and outer radius $5\,\text{m}$, to 1 decimal place?

Large circle area = $\pi \times 5^2 = 25\pi \approx 78.5\,\text{m}^2$; Shaded area = $25\pi - 9\pi = 16\pi \approx 50.3\,\text{m}^2$

In Page 34, Question 6, a square with side length $8\,\text{cm}$ has all four vertices on a circle. What are the diagonal of the square and the shaded area outside the square?

Diagonal = $\sqrt{8^2 + 8^2} = \sqrt{128} \approx 11.3\,\text{cm}$; Shaded area = $\pi \times \left(\frac{\sqrt{128}}{2}\right)^2 - 8^2 = 32\pi - 64 \approx 36.5\,\text{cm}^2$

In Source 12, formula (a), make $x$ the subject: $y = \frac{ax}{b} + c$

$$x = \frac{b(y - c)}{a}$$

In Source 12, formula (b), make $x$ the subject: $y = \frac{ax^3}{b}$

$$x = \sqrt[3]{\frac{by}{a}}$$

In Source 12, formula (c), make $x$ the subject: $ax + by = c$

$$x = \frac{c - by}{a}$$

In Source 12, formula (d), make $x$ the subject: $y = \sqrt{\frac{bx}{10}}$

$$x = \frac{10y^2}{b}$$

In Source 12, formula (e), make $x$ the subject: $y + a = \frac{x}{2b} - 3a$

$$x = 2b(y + 4a)$$

In Source 12, formula (f), make $x$ the subject: $x^2 + y^2 = 9$

$x = \pm\sqrt{9 - y^2}$ (or $x = \sqrt{9 - y^2}$)

In Source 12, formula (g), make $x$ the subject: $y = ab + \sqrt{cx}$

$$x = \frac{(y - ab)^2}{c}$$

In Source 12, formula (h), make $x$ the subject: $y = \frac{2x + a}{b}$

$$x = \frac{by - a}{2}$$

In Source 12, formula (i), make $x$ the subject: $y = \sqrt[3]{5x - a}$

$$x = \frac{y^3 + a}{5}$$

In Source 12, formula (j), make $x$ the subject: $y = \frac{a(x - 2)}{b}$

$$x = \frac{by}{a} + 2$$

In Source 12, formula (k), make $x$ the subject: $y + 6 = \frac{ax}{2} - 4y$

$$x = \frac{2(5y + 6)}{a}$$

In Source 12, formula (l), make $x$ the subject: $y^3 = \frac{x^2}{a} - b^2$

$x = \pm\sqrt{a(y^3 + b^2)}$ (or $x = \sqrt{a(y^3 + b^2)}$)