L1 Mathematical Reasoning
Question Bank
The perimeter of the rectangle shown below is 76 cm.
Form an equation and solve it to find the value of $x$.
The length of a rectangular garden is 4 metres longer than its width. The total area of the garden is 45 m².
By forming and solving a quadratic equation, find the dimensions of the garden.
The right-angled triangle shown has an area of 21 cm². Calculate the value of x.
The diagram below shows the design for a custom metal sign. The design starts with a 50 cm by 30 cm rectangular sheet of metal, and four identical squares are cut from the corners. The final area of the sign is 1400 cm². Find the side length, $x$, of the squares that were removed.
The semi-circle shown in the diagram below has an area of 40 cm². Calculate the value of y.
Alex is building a new wooden deck in their backyard. The shape of the deck is a trapezium, with the dimensions shown in the diagram below.
To order the right amount of wood and stain, Alex needs to calculate the total area of the deck. Calculate the area of the deck.
The square shown in the diagram has an area of 40 cm².
Calculate the length of the diagonal of the square.
A designer is creating a mirror in the shape of a regular hexagon. The perpendicular distance from the center to the middle of any side is 40 cm.
(i) Achieved Show that the size of the interior angle, marked 'x', is 120°.
(ii) Merit Find the total area of the mirror.
A landscaper is building a patio in the shape of a regular pentagon. The length of one side of the patio is 3 metres.
(i) Achieved Calculate the size of angle 'v', which is the angle between the line from a corner to the center and the adjacent side.
(ii) Merit Find the total area of the patio.
The diagram shows a rhombus with a perimeter of 80 cm. The length of the diagonal that does not contain angle $x$ is 24 cm.
Find the size of angle $x$. Show your working clearly.
A rhombus has a total area of 96 cm². One of its diagonals measures 16 cm.
Find the size of the smaller interior angle of the rhombus.
An isosceles trapezium has parallel sides of length 10 cm and 22 cm. The two non-parallel (slanted) sides are each 10 cm long. Find the area of the trapezium.
A right-angled triangle has side lengths in the ratio 3 : 4 : 5. If the area of the triangle is 216 cm², find the perimeter of the shape.
The shape shown is a hexagram, made up of 12 identical equilateral triangles. The side length of each of these small triangles is 5 cm.
(i) Achieved Find the perimeter of the hexagram.
(ii) Merit Find the total area of the hexagram.
The diagram below shows the top of a table which is in the shape of a regular octagon. The distance from the center to any corner is 120 cm.
Find the area of the octagon. You must show your working clearly.
The right-angled triangle and the square shown in the diagram have the same area. Calculate the value of x.
The length of a rectangle is 6 cm longer than the width. If the length is increased by 9 cm and the width by 5 cm, the new perimeter will be 160 cm. Find the dimensions of the original rectangle.
The International Space Station (ISS) orbits at an altitude of 200 km above the Earth's surface. The total distance it travels in one complete orbit is 42,250 km. As shown in the diagram, the radius of the Earth is marked as $r$. Calculate the radius of the Earth.
The perimeter of the semi-circle shown in the diagram is 30 m. The radius is labelled as y. Calculate the value of y giving your answer in terms of $\pi$.
An artist is designing a triangular stained-glass window. The design consists of a large outer equilateral triangle containing a smaller blue equilateral triangle. As shown in the diagram, the space between them is made up of three identical red right-angled triangles.
The side lengths of the large triangle are 3 cm. Each of the red triangles has a short side of 1 cm and a hypotenuse of 2 cm.
Calculate the ratio of the total area of the red glass to the area of the blue glass. Show your working.
A graphic designer creates a logo by inscribing a circle within a rectangle PQRS, as shown in the diagram. The side PS is a diameter of the circle. Let the radius of the circle be $r$.
(a) Calculate the ratio of the perimeter of the rectangle to the circumference of the circle.
(b) Calculate the ratio of the area of the rectangle to the area of the circle.
Give your answers in their simplest form, in terms of $\pi$.
The diagram below shows a regular hexagon. The distance from the center of the hexagon to any corner is $d$.
(i) Find an expression for the perimeter of the hexagon in terms of $d$.
(ii) Find an expression for the area of the hexagon in terms of $d$.
The diagram shows a regular octagon. The distance from the center to any corner is $d$.
(i) Find an expression for the perimeter of the octagon in terms of $d$.
(ii) Find an expression for the area of the octagon in terms of $d$.
The area of the right-angled trapezium shown below is 15 cm². Find the perimeter of the shape.
Four identical parallelograms make a pinwheel shape, as shown in the diagram. The area of the entire shape is 84 m². Find the perimeter of the shape.
The sides of an isosceles triangle are in the ratio 8 : 5 : 5. If the area of the triangle is 84 cm², what is the perimeter?
An equilateral triangle has a base of $2x$. Find an expression for its area in terms of $x$.
Both a square and a rectangle each have a perimeter of 44 cm. If their combined area is 238 cm², what are the dimensions of the rectangle?
A regular hexagon is divided into two rectangles and four triangles as shown. The distance $k$ is the height of the triangles.
(i) Merit Find an expression for the perimeter of the hexagon in terms of $k$.
(ii) Excellence Find an expression for the area of the hexagon in terms of $k$.
A rectangular flower bed is surrounded by a paved path, as shown in the diagram below. All measurements are in metres.
Find a simplified expression for the area of the paved path.
The diagram shows part of a regular polygon with an interior angle of 144° and a side length of 5 cm. Calculate the total area of the polygon.
A regular octagon is divded up by drawing lines from each vertex to the 'opposite' vertex, and a length is marked $d$, as shown in the diagram below.
Find an expression for the total area of the octagon in terms of $d$.
A hardware store is bundling 16 identical pipes for sale. The pipes are arranged in a 4x4 grid, and a metal band is wrapped tightly and securely around the outside of the bundle.
Each pipe has an inner radius of $r$ cm and a wall thickness of 2 cm. Find a expression for the total length of the metal band in terms of $r$ and $\pi$, give your answer in fully factorised form.