GCSE / IGCSE Maths – Geometry

(a) Work out the size of each interior angle of a regular decagon.(b) A regular polygon has interior angles of $156^\circ$. Work out the number of sides.(c) Two parallel lines are crossed by a transversal. One of the angles formed is $68^\circ$. Write down the size of the co-interior (allied) angle to it, giving a reason.

(a) A right-angled triangle has shorter sides of length $7\,\text{cm}$ and $24\,\text{cm}$. Work out the length of the hypotenuse.(b) A cuboid measures $3\,\text{cm}$ by $4\,\text{cm}$ by $12\,\text{cm}$. Work out the length of the longest diagonal of the cuboid.

(a) A trapezium has parallel sides of length $8\,\text{cm}$ and $14\,\text{cm}$ and a perpendicular height of $5\,\text{cm}$. Work out its area.(b) A sector of a circle has radius $6\,\text{cm}$ and angle $60^\circ$. Work out the exact area of the sector and the exact arc length, leaving your answers in terms of $\pi$.

A cone has base radius $3\,\text{cm}$ and perpendicular height $4\,\text{cm}$.(a) Show that the slant height is $5\,\text{cm}$.(b) Work out the volume of the cone, giving your answer in terms of $\pi$.(c) Work out the curved surface area, giving your answer in terms of $\pi$. (Curved surface area of a cone $= \pi r l$.)

The point $P$ has coordinates $(2, 3)$.(a) Write down the coordinates of the image of $P$ after a rotation of $90^\circ$ clockwise about the origin.(b) Write down the coordinates of the image of $P$ after a reflection in the line $y = x$.(c) The point $Q(3, 2)$ is enlarged by scale factor $-2$ with centre $(1, 1)$. Find the coordinates of the image of $Q$.(d) Describe fully the single transformation that maps triangle $A$ to triangle $B$, where $B$ is the same shape and size as $A$, is the same way round (not a mirror image), and every point has moved $5$ units right and $2$ units down.

Points $A$, $B$ and $C$ lie on a circle with centre $O$. Angle $AOB = 110^\circ$ and $C$ is on the major arc.(a) Work out angle $ACB$, giving a reason.(b) $D$ is a point on the minor arc $AB$. Work out angle $ADB$, giving a reason.(c) A tangent to the circle at $A$ meets the line $OA$. Write down the angle between the tangent and $OA$.

A cuboid $ABCDEFGH$ has base $ABCD$ with $AB = 6\,\text{cm}$ and $BC = 8\,\text{cm}$, and vertical height $CG = 5\,\text{cm}$. $G$ is vertically above $C$.(a) Work out the length $AC$.(b) Work out the length $AG$, giving your answer to 3 significant figures.(c) Work out the angle that $AG$ makes with the base $ABCD$, giving your answer to 1 decimal place.

A solid is made from a cylinder of radius $3\,\text{cm}$ and height $8\,\text{cm}$ with a hemisphere of radius $3\,\text{cm}$ fixed on top.(a) Work out the volume of the solid, giving your answer in terms of $\pi$.(b) The solid is made of metal with density $8.9\,\text{g/cm}^3$. Work out the mass of the solid to the nearest gram. (Volume of a sphere $= \dfrac{4}{3}\pi r^3$.)

Two mathematically similar bottles have heights $12\,\text{cm}$ and $18\,\text{cm}$.(a) The label on the smaller bottle has area $30\,\text{cm}^2$. Work out the area of the corresponding label on the larger bottle.(b) The larger bottle holds $540\,\text{ml}$. Work out how much the smaller bottle holds.

$A$, $B$ and $C$ are points on a circle with centre $O$. The line $TA$ is a tangent to the circle at $A$. Angle $OAB = 35^\circ$.(a) Work out angle $AOB$, giving a reason.(b) Work out angle $ACB$, where $C$ is on the major arc, giving a reason.(c) Work out angle $TAB$ (the angle between the tangent and the chord $AB$) and state the circle theorem that connects your answers to (b) and (c).

A shape is made from a square of side $10\,\text{cm}$ with a quarter circle of radius $5\,\text{cm}$ removed from each of two adjacent corners (the quarter circles are centred at the two corners and do not overlap).(a) Work out the area of the shape to 3 significant figures.(b) Work out the perimeter of the shape to 3 significant figures.

A solid cone has base radius $6\,\text{cm}$ and height $12\,\text{cm}$. The top of the cone is removed by a cut parallel to the base, $4\,\text{cm}$ from the vertex, leaving a frustum.(a) Show that the radius of the circular cut is $2\,\text{cm}$.(b) Work out the volume of the frustum, giving your answer in terms of $\pi$.(c) The frustum is melted down and recast as a sphere. Work out the radius of the sphere to 3 significant figures.

Triangle $A$ has vertices $(1, 1)$, $(3, 1)$ and $(1, 4)$.(a) Triangle $A$ is reflected in the $x$-axis to give triangle $B$. Write down the vertices of $B$.(b) Triangle $B$ is then rotated $180^\circ$ about the origin to give triangle $C$. Write down the vertices of $C$.(c) Describe fully the single transformation that maps triangle $A$ directly to triangle $C$.(d) Triangle $A$ is enlarged by scale factor $3$ about the origin. Work out the area of the enlarged triangle.