Exercises: Pythagoras' Theorem

A fire engine ladder rests against the façade of a building. The base of the ladder is $3\text{ m}$ from the wall and reaches a height of $4\text{ m}$ on the wall. Calculate the total length of the ladder.

A telecommunications mast is held by a cable tensioned from its top end to a point on the ground. The cable measures $13\text{ m}$ and the mast has a height of $12\text{ m}$. Determine how far from the base of the mast the cable is anchored to the ground.

The dimensions of a rectangular computer screen are $32\text{ cm}$ wide and $24\text{ cm}$ high. Calculate the length of its diagonal to find the actual screen size.

A town square is in the shape of a square with side length $50\text{ m}$. If you decide to cross it in a straight line from one corner to the opposite corner, what distance will you cover?

To build a rhombus-shaped kite, two wooden rods measuring $60\text{ cm}$ and $80\text{ cm}$ have been crossed. Calculate the length of the outer edge of the kite (the side of the rhombus).

A road sign is in the shape of an isosceles triangle. Its base measures $60\text{ cm}$ and each of its two equal sides measures $50\text{ cm}$. Calculate the height $h$ of the sign.

In the design of an industrial warehouse roof, the main frame forms a right-angled triangle. The altitude to the hypotenuse divides it into two segments of $4\text{ m}$ and $9\text{ m}$. Apply the altitude theorem to calculate the length of this altitude.

In a park, a flower bed is in the shape of a right-angled triangle. The hypotenuse measures $25\text{ m}$ and the projection of one of the legs onto the hypotenuse measures $9\text{ m}$. Use the leg theorem to calculate the length of that leg.

The cross-section of an irrigation canal is in the shape of an isosceles trapezium. The width at the top is $8\text{ m}$, the width at the bottom is $2\text{ m}$ and the depth of the canal is $4\text{ m}$. Calculate the length of the sloping sides $x$ of the canal.

A decorative mosaic is made of regular hexagonal tiles with a side length of $10\text{ cm}$. Calculate the apothem of one of these tiles.

A plot of land is in the shape of a right-angled trapezium. The side perpendicular to the parallel bases measures $12\text{ m}$, the longer base measures $30\text{ m}$ and the shorter base measures $25\text{ m}$. Calculate the total perimeter of the plot.

A cylindrical pipe carries water. The radius of the pipe is $13\text{ cm}$ and the surface of the water forms a horizontal chord $24\text{ cm}$ wide. If the water level is below the center of the pipe, what is the maximum depth of the water $h$?

The roof of a house is in the shape of a right-angled triangle resting on its hypotenuse. The height of the roof relative to the horizontal base is $6\text{ m}$ and the projection of one of the roof slopes (leg) onto the base is $8\text{ m}$. Calculate the length of the other projection and the lengths of the two sloping sides of the roof.

To transport a rigid metal bar, a rectangular prism-shaped box with dimensions $80\text{ cm}$ long, $60\text{ cm}$ wide, and $50\text{ cm}$ high is available. Calculate the maximum length the bar can have so that it fits completely inside the closed box.

A vertical aerial is held by two cables anchored to the ground on opposite sides of the base. One cable measures $15\text{ m}$ and is anchored $9\text{ m}$ from the base. The other cable is anchored $16\text{ m}$ from the base. Calculate the height of the aerial and the length of the second cable.

The perimeter of a rhombus-shaped swimming pool is $52\text{ m}$ and one of its diagonals measures $10\text{ m}$. Calculate the length of the other diagonal and the surface area of the pool.

The Great Pyramid of Giza has a square base. Suppose the side of its base measures $230\text{ m}$ and the length of one of its side edges (from a base vertex to the apex) is $219\text{ m}$. Calculate the height $h$ of the pyramid.

In a garden, a section in the shape of a right-angled triangle is to be enclosed. It is known that the lengths of its three sides are three consecutive even integers. Set up the corresponding equation using Pythagoras' Theorem and determine the lengths of all three sides.

In a circular piece of wood with a radius of $25\text{ cm}$, two straight parallel cuts are made. One cut has a length of $48\text{ cm}$ and the other $40\text{ cm}$. If both cuts are made on opposite sides of the center of the circle, calculate the perpendicular distance separating the two cuts.

A room is in the shape of a rectangular prism with dimensions $5\text{ m}$ long, $4\text{ m}$ wide and $3\text{ m}$ high. A spider is in one corner of the ceiling and wants to reach the opposite corner on the floor by crawling along the walls, ceiling or floor. If the spider takes the shortest possible path, calculate this distance. (Hint: consider the net of the room).

In a right-angled triangle, the altitude to the hypotenuse measures $12\text{ cm}$ and one of the legs measures $15\text{ cm}$. Using the leg theorem and altitude theorem, or Pythagoras' Theorem iteratively, calculate the length of the hypotenuse $a$ and the other leg $c$.

A bridge has a side frame in the shape of a scalene trapezium. The horizontal parallel bases measure $40\text{ m}$ and $26\text{ m}$, and the non-parallel sides measure $13\text{ m}$ and $15\text{ m}$. Set up a system of equations using Pythagoras' Theorem to calculate the height of the trapezium.

A logo is formed by an isosceles triangle inscribed in a circle. The base of the triangle measures $16\text{ cm}$ and its height relative to the base measures $18\text{ cm}$. Calculate the radius of the circle in which the triangle is inscribed.

Two towers of height $15\text{ m}$ and $20\text{ m}$ are separated by an unknown horizontal distance. A bird flies in a straight line from the top of the $15\text{ m}$ tower to the base of the $20\text{ m}$ tower, travelling exactly $25\text{ m}$. Calculate the distance between the bases of the towers. Then, if another bird flies in a straight line from the top of the $20\text{ m}$ tower to the base of the $15\text{ m}$ tower, what distance will it travel?