Grade 7 Mathematics Examination: Geometry, Measurement, and Problem Solving Study Guide

Area of a Rectangle: The area ( $A$ ) is calculated by multiplying the length ( $L$ ) by the breadth ( $B$ ). - Formula: $A = L \times B$ - Application Example: If a rectangle has an area of $45\,cm^2$ and a length of $9\,cm$ , the breadth is calculated as $\frac{45\,cm^2}{9\,cm} = 5\,cm$ .
Area of a Triangle: The area is calculated as half the product of the base ( $b$ ) and the perpendicular height ( $h$ ). - Formula: $A = \frac{1}{2} (b \times h)$
Perimeter of a Square: The perimeter ( $P$ ) is the total distance around the square, calculated by multiplying the length of one side ( $s$ ) by 4. - Formula: $P = 4 \times s$ - Application Example: If the perimeter of a square is $24\,cm$ , the length of one side is $\frac{24\,cm}{4} = 6\,cm$ .
Sum of Interior Angles in a Quadrilateral: The sum of the interior angles of any quadrilateral is always equal to $360^{\circ}$ .
Sum of Interior Angles in a Triangle: The sum of the interior angles of any triangle is always equal to $180^{\circ}$ .

Polygons by Side Count: - A polygon with nine sides is classified as a nonagon.
Triangle Classifications: - Equilateral Triangle: A triangle in which all three sides are of equal length and all internal angles are equal ( $60^{\circ}$ ). - Isosceles Triangle: A triangle with at least two equal sides and two equal angles. - Scalene Triangle: A triangle where all three sides and all three angles have different measurements. - Right-angled Triangle: A triangle containing one angle that measures exactly $90^{\circ}$ .

Straight Angle: An angle that measures exactly $180^{\circ}$ , forming a straight line.
Angle Notation and Measurement: - Angles are named using three letters, such as $\angle PQR$ , where the middle letter represents the vertex. - Example: To construct and label an angle $\angle PQR$ measuring $50^{\circ}$ , one uses a protractor to ensure high precision at the vertex $Q$ .
Calculation of Missing Angles: Using the principle that interior angles of a triangle sum to $180^{\circ}$ , the missing angle $x$ can be found using: - $x = 180^{\circ} - (\text{angle}_1 + \text{angle}_2)$

Circumference: The outline, perimeter, or border around the outside of a circle.
Radius ( $r$ ): The distance from the center of the circle to any point on its circumference.
Diameter ( $d$ ): The distance across the circle passing through the center; it is twice the length of the radius ( $d = 2r$ ).
Concentric Circles: Two or more circles that share the same center point (Centre Point A) but have different radii. - Example: A circle with a diameter of $100\,mm$ (radius of $50\,mm$ or $5\,cm$ ) drawn around a circle with a radius of $3\,cm$ .
Chord: A straight line segment whose endpoints both lie on the circumference of a circle. - Specific Task Requirements: Drawing a chord ( $DE$ ) in a larger concentric circle such that it does not intersect or touch the circumference of the smaller inner circle.

Volume of a Rectangular Prism (CD Box): Volume ( $V$ ) is the measure of the space inside a 3D object, calculated by multiplying length, breadth, and height. - Formula: $V = L \times B \times H$ - Calculation: For a box with height $2\,cm$ , length $14\,cm$ , and breadth $12\,cm$ : - $V = 14\,cm \times 12\,cm \times 2\,cm = 336\,cm^3$
Surface Area of a Rectangular Prism: The total area of all the faces of the 3D shape. - Formula: $SA = 2(L \times B) + 2(L \times H) + 2(B \times H)$
Anatomy of 3D Shapes: - Faces: The flat surfaces of a 3D solid. - Edges: The line segments where two faces meet. - Vertices: The corner points where three or more edges meet.

Compound Area (Shaded Regions): - To find the area of a shaded region, one typically calculates the area of the outer/larger shape and subtracts the area of the inner/smaller shape. - $\text{Area}<em>{\text{Shaded}} = \text{Area}</em>{\text{Total}} - \text{Area}_{\text{Unshaded}}$
Fencing and Guardrail Problems (Perimeter Application): - Scenario: A rectangular garden measuring $14\,m$ long and $7\,m$ wide needs a fence, excluding a $2.5\,m$ opening for a gate. - Full Perimeter Calculation: $P = 2(L + B) = 2(14\,m + 7\,m) = 2(21\,m) = 42\,m$ - Fence Length: $\text{Total Perimeter} - \text{Gate Opening} = 42\,m - 2.5\,m = 39.5\,m$ - Cost Analysis: - At $R47$ per metre: $39.5\,m \times R47/m = R1\,856.50$ - At $R39$ per metre: $39.5\,m \times R39/m = R1\,540.50$ - Total Savings: $R1\,856.50 - R1\,540.50 = R316.00$