Geometry and Trigonometry

In a scalene triangle, how many sides are equal in length?

What is the length of the arc of a semi-circle with a radius of $7\text{ cm}$? (Use $\pi \approx \frac{22}{7}$)

Find the volume of a square-based pyramid where the base side is $3\text{ cm}$ and the height is $4\text{ cm}$. (Use $V = \frac{1}{3}Bh$)

How many sides does a heptagon have?

In a triangle, the sides are $5\text{ cm}$, $12\text{ cm}$, and $13\text{ cm}$. What type of triangle is it?

Calculate the surface area of a rectangular prism with dimensions $3\text{ cm}$ by $4\text{ cm}$ by $5\text{ cm}$.

The area of a square is $64\text{ cm}^2$. What is the length of its diagonal?

A sphere is placed inside a cylinder with the same radius $r$ and height $2r$. What fraction of the cylinder's volume is occupied by the sphere?

Find the area of a sector whose perimeter is $20\text{ cm}$ and radius is $5\text{ cm}$.

A right square pyramid has a volume of $48\text{ cm}^3$ and a base area of $36\text{ cm}^2$. What is the length of its slant edge?

In a right-angled triangle, the legs are $60\text{ mm}$ and $80\text{ mm}$. What is the length of the hypotenuse?

A rhombus has diagonals of $12\text{ cm}$ and $16\text{ cm}$. What is the length of its side?

The hypotenuse of a right-angled triangle is $1.7\text{ m}$ and one leg is $0.8\text{ m}$. Find the other leg.

A right-angled triangle has legs of $1.6\text{ cm}$ and $3\text{ cm}$. Find the hypotenuse.

A triangle has side lengths of $9\text{ cm}$, $12\text{ cm}$, and $x\text{ cm}$. If the triangle is right-angled and $x$ is the hypotenuse, what is the value of $x$?

Find the length of the side of a square whose diagonal is $12\text{ cm}$. (Give your answer to two decimal places)

A person walks $12\text{ m}$ North and $5\text{ m}$ East, and then climbs a vertical ladder of $8.4\text{ m}$. How far is the person from the starting point on the ground?

A regular square pyramid has a base area of $144\text{ cm}^2$ and a slant height of $10\text{ cm}$. Find the vertical height of the pyramid.

In a coordinate plane, what is the distance between $(-5, 2)$ and $(7, -3)$?

Find the area of a right triangle where the hypotenuse is $25\text{ cm}$ and one leg is $7\text{ cm}$.

Find the midpoint of the points $(10, 20)$ and $(30, 40)$.

Find the distance between the point $(3, 4)$ and the origin.

Calculate the gradient of the line $y - 3 = 2(x - 1)$.

What is the midpoint of the segment connecting $(a, b)$ and $(3a, 5b)$?

Find the midpoint of the segment between $(-10, 5)$ and $(4, -3)$.

Calculate the gradient of the line segment between the points $(2, 5)$ and $(2, -5)$.

What is the equation of the line passing through the origin with a gradient of $-1$?

The point $P(3, 4)$ is reflected over the line $x+y=10$ to point $P'$. Find the coordinates of $P'$.

If the point $(k, k+1)$ lies on the perpendicular bisector of the segment joining $(1, 2)$ and $(5, 4)$, find $k$.

Find the coordinates of the point $P$ on the $x$-axis such that the sum of the squares of its distances from $(0, 3)$ and $(4, 0)$ is minimized.

In $\triangle PQR$, $\angle Q = 90^\circ$, $PR = 10$, and $\angle P = 50^\circ$. What is the length of side $QR$?

If $\cos x = \frac{1}{\sqrt{2}}$ and $x$ is acute, find $x$.

In $\triangle ABC$, $\angle B = 90^\circ$, $BC = 6$, and $AB = 8$. What is the value of $\tan A$?

Find the height of a tree if its shadow is $15$ m long when the angle of elevation of the sun is $25^\circ$.

A boat sails $20$ km South and then $12$ km West. What angle does the path of the boat make with the South direction?

In a right-angled triangle, one of the acute angles is $38^\circ$. If the side adjacent to this angle is $18$ cm, find the length of the hypotenuse to two decimal places.

A kite is flying at a height of $55$ m above the ground. If the length of the string is $100$ m, find the angle the string makes with the ground.

Given $\sin \theta = ρ$, where $0 < ρ < 1$ and $\theta$ is acute, find $\tan(90 - \theta)$ in terms of $ρ$.

A path goes straight up a hill. The path makes an angle of $15^\circ$ with the horizontal. If you walk $100$ meters along the path, your bearing changes from $000^\circ$ to $010^\circ$ due to the curvature of the hill (projected onto a map). What is the horizontal distance traveled?

A pyramid has a rectangular base $10$ cm by $8$ cm. The vertex is directly above the center of the base and is $12$ cm high. Calculate the angle of elevation of an edge from a corner of the base.

A pyramid has a base area of $21$ cm$^2$ and a height of $10$ cm. What is its volume?

A rectangular prism has dimensions $2$ cm by $8$ cm by $5$ cm. What is its volume?

What is the volume of a cube with side length $1$ meter?

Find the curved surface area of a hemisphere with a radius of $3$ cm in terms of $\pi$.

A rectangular box has dimensions $a$, $2a$, and $3a$. If its total surface area is $88$ cm$^2$, find the value of $a$.

A hollow cylinder has an outer radius of $2$ cm and an inner radius of $1$ cm. If its height is $10$ cm, find the volume of the material in terms of $\pi$.

Two cubes have side lengths of $2$ cm and $3$ cm respectively. Find the ratio of their total surface areas.

A composite solid is formed by a cylinder with radius $r$ and height $h$ and two identical cones attached to each end. Each cone has radius $r$ and height $h$. Find the total volume of the solid.

A sphere of radius $r$ is placed inside a cylinder of radius $r$ and height $2r$. What fraction of the cylinder's volume is empty space?

Find the volume of the largest cone that can be carved out of a cube of side $6$ cm.

What is the scale factor of an enlargement centered at the origin that maps point $(2, 2)$ to $(8, 8)$?

Point $S(1, 4)$ is reflected in the line $y = x$. What are the coordinates of the image $S'$?

Point $R(5, 5)$ is rotated $90^\circ$ clockwise about the origin. What are the coordinates of the image $R'$?

Point $Q(3, -2)$ is translated by the vector $\binom{0}{5}$. What are the coordinates of the image $Q'$?

An enlargement with scale factor $k = -1$ with the origin as the center is equivalent to which other transformation?

A translation moves a point 3 units to the left and 4 units up. What is the translation vector?

Point $D(1, 0)$ is rotated $90^\circ$ counter-clockwise about the origin, and then the image is reflected in the $x$-axis. What is the final point?

Point $L(0, 5)$ is rotated $90^\circ$ anticlockwise about $(2, 2)$, then reflected in $y = x$. The final image is:

An enlargement maps $(2, 3)$ to $(4, 6)$ and $(3, 3)$ to $(7, 6)$. Find the center of enlargement.

Point $S(2, -4)$ is translated by $\binom{-1}{3}$ and then reflected in the line $x = 0$. Find $S''$.

Convert $150$ millimeters to centimeters.

The area of a park is $2.5\text{ km}^2$. How many square meters ($m^2$) is this?

Convert $0.1$ kilograms to grams.

A piece of wood is $7.2$ meters long. How many centimeters is this?

A rectangle has an area of $48 \text{ cm}^2$. If the length is $0.08 \text{ m}$, what is the width in $mm$?

A fence is $0.012 \text{ km}$ long. If fence posts are placed every $80 \text{ cm}$, how many posts are needed (including both ends)?

How many $250 \text{ mL}$ cups are needed to fill a container with a volume of $0.02 \text{ m}^3$?

A tank in the shape of a cube with edge $2 \text{ m}$ is half-full of water. How many more $liters$ of water are needed to fill it to $80\%$ of its capacity?

Convert $0.055 \text{ decimetres}^2$ ($dm^2$) to $mm^2$.

A cylindrical wire is $100 \text{ m}$ long and has a radius of $1 \text{ mm}$. Find the mass of the wire in grams if the density of the metal is $8 \text{ g/cm}^3$. (Use $\pi \approx 3.14$)

If a square with side length 3 is enlarged by a scale factor of 4, what is the side length of the new square?

Two triangles are congruent if two angles and a non-included side of one are equal to the corresponding parts of the other. This is the:

If the ratio of the areas of two similar circles is $1:100$, what is the ratio of their radii?

A map has a scale of $1:10000$. If a road is 5 cm on the map, how long is it in reality?

Two cylinders are similar with heights of $3 \text{ cm}$ and $6 \text{ cm}$. If the surface area of the smaller cylinder is $20 \text{ cm}^2$, what is the surface area of the larger cylinder?

In $\triangle ABC$, $DE \parallel BC$. If $AD/AB = 1/3$, what is the ratio of the area of $\triangle ADE$ to the area of $\triangle ABC$?

The ratio of the areas of two similar triangles is $16:25$. If the perimeter of the larger triangle is $50 \text{ cm}$, what is the perimeter of the smaller triangle?

A solid metal cube is melted and recast into 27 smaller identical cubes. What is the ratio of the surface area of the original cube to the sum of the surface areas of the 27 smaller cubes?

In $\triangle ABC$, $D$ is on $AB$ and $E$ is on $AC$ such that $DE \parallel BC$. If $AD = 2\text{ cm}$, $AB = 5\text{ cm}$, and the area of trapezoid $DBCE$ is $63\text{ cm}^2$, find the area of $\triangle ADE$.

Two similar pentagons have perimeters in the ratio $4:9$. If the area of the smaller pentagon is $32 \text{ cm}^2$, find the area of the larger pentagon in $\text{cm}^2$.

In a circle, any angle subtended by a diameter at the circumference is:

What is the area of a circle with a radius of $10\text{ cm}$? Use $\pi \approx 3.14$.

What is the central angle of a sector that represents $\frac{1}{8}$ of a circle?

The area of a circle is $154\text{ cm}^2$. Using $\pi \approx \frac{22}{7}$, find the radius.

A chord of length $30$ cm is $8$ cm from the center of the circle. Find the diameter of the circle.

In a circle with center $O$, the major arc $AB$ subtends an angle of $250^\circ$ at the center. Find the angle subtended by the minor arc $AB$ at the circumference.

Find the circumference of a circle in which an arc of $90^\circ$ has a length of $12$ cm.

A circle is inscribed in an isosceles trapezoid with bases $16$ cm and $36$ cm. Find the radius of the circle.

In a cyclic quadrilateral $ABCD$, $\angle DAB = 110^\circ$. If $AD \parallel BC$, find the measure of $\angle ABC$.

In a circle, $PT$ is a tangent at $T$ and $PAB$ is a secant. If $PT = 12$ cm and $PA = 8$ cm, find the length of chord $AB$.

A plane is flying on a bearing of $310^\circ$. It then turns $20^\circ$ to its left. What is its new bearing?

A ship is sailing on a bearing of $080^\circ$. It then turns $40^\circ$ to its right. What is its new bearing?

If a town $Q$ is on a bearing of $220^\circ$ from town $P$, what is the bearing of $P$ from $Q$?

Calculate the back bearing of $295^\circ$.

A ship travels on a bearing of $200^{\circ}$. In which compass quadrant is the ship traveling?

The bearing of town $M$ from town $N$ is $115^{\circ}$. Calculate the angle that the line $NM$ makes with the East direction, measured South of East.

A drone flies $10$ km North, $10$ km East, and $10$ km South. What is its bearing from the starting point?