Mensuration

The area of a regular hexagon is $54\sqrt{3} \text{ cm}^2$. Find the length of one side.

A rhombus has diagonals of length $12$ cm and $16$ cm. Find its perimeter.

Find the area of the region bounded by the curve $y = 4 - x^2$ and the $x$-axis.

Find the area of a trapezoid with parallel sides of length $10$ cm and $16$ cm, and non-parallel sides of length $5$ cm each.

A garden path is designed in the shape of a circular sector with radius $r$ and angle $\theta$ (in radians). It is known that the perimeter of the sector is exactly $20$ meters. Let $A$ be the area of this sector. Find the value of $r$ that maximizes the area $A$.

A rectangle is inscribed in a semi-circle of radius $R$ such that two vertices lie on the diameter and two on the arc. Find the maximum area of such a rectangle.

A circular pond has a radius of $10$ m. A bridge is built across the pond in the shape of a chord that is $12$ m long. Find the area of the smaller part of the pond divided by the bridge.

A logo consists of a large equilateral triangle of side $s = 6$ cm. Within this triangle, three identical circles are drawn such that each circle is tangent to two sides of the triangle and to the other two circles. Find the area of the region in the center of the triangle not covered by the circles.

An ellipse is defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The area is given by $\pi ab$. A rectangle is inscribed in the ellipse with sides parallel to the axes. Find the maximum possible area of such a rectangle in terms of $a$ and $b$.

A piece of wire $40$ cm long is cut into two pieces. One piece is bent to form a square and the other to form a circle. If the sum of the areas is a minimum, find the radius of the circle.

What is the area of a semicircle with radius $10$ cm? (Use $\pi$ in your answer)

A wire of length $20$ cm is bent into the shape of a sector of a circle with radius $6$ cm. Find the angle of the sector in radians.

Convert an angle of $45^{\circ}$ to radians.

In a circle of radius $9$ cm, a sector has an area of $27$ cm$^2$. Find the length of the arc of this sector.

A metal plate is in the shape of a sector $OPQ$ of a circle with center $O$ and radius $12$ cm. The area of the sector is $108$ cm$^2$. A smaller sector $ORS$ with center $O$ and radius $8$ cm is removed from the plate. Calculate the length of the arc $RS$.

The diagram shows a sector $OAB$ of a circle with center $O$ and radius $r = 8$ cm. The angle $AOB$ is $\theta$ radians. The perimeter of the sector is $20$ cm. Find the area of the sector $OAB$.

The perimeter of a certain sector is $8$ times its radius. What is the area of the sector in terms of its radius $r$?

A wiper on a car's windshield rotates through an angle of $120^\circ$. The wiper blade is $40 \text{ cm}$ long and is attached to an arm such that the end of the blade nearest the pivot is $15 \text{ cm}$ from the pivot. Find the area of the windshield cleaned by the blade, to the nearest whole number.

An equilateral triangle $ABC$ has side length $10 \text{ cm}$. Three circular arcs are drawn with centers $A$, $B$, and $C$, each with radius $5 \text{ cm}$, such that they meet at the midpoints of the sides. Calculate the area of the region inside the triangle but outside the three sectors.

A running track consists of two parallel straight sections, each $100 \text{ m}$ long, and two semicircular ends. The inner radius of the semicircular ends is $30 \text{ m}$ and the track is $5 \text{ m}$ wide. Find the total area of the track surface.

A cylinder has a volume of $100\pi$ cm$^3$ and a height of $4$ cm. What is its radius?

A right-angled triangular prism has a length of $12$ cm. The triangular base has legs of $3$ cm and $4$ cm. What is its volume?

The total surface area of a cube is $150$ cm$^2$. What is the length of one side of the cube?

A prism has a cross-sectional area of $25$ cm$^2$ and a volume of $250$ cm$^3$. What is the length of the prism?

A right cone is cut parallel to its base at half its vertical height. What is the ratio of the volume of the small top cone to the volume of the original cone?

A container is a hemisphere of radius $r$ surmounted by a cylinder of the same radius and height $2r$. Find the total volume of the container.

The total surface area of a cylinder is $600\pi$ cm$^2$. If the height is $20$ cm, find the radius.

A container is in the shape of an inverted right circular cone with height $12$ cm and base radius $6$ cm. Water is being poured into the container at a constant rate. When the depth of water is $h$, the surface area of the water (the top circular part) is $A$. Find $dA/dh$ when $h = 4$ cm.

A cylindrical log of radius $R$ and length $L$ is carved to produce the largest possible square prism. Find the volume of the wood removed during this process.

A solid consists of a cone of height $h$ and base radius $r$ fixed to the base of a hemisphere of radius $r$. If the total volume is $V = 100\pi \text{ cm}^3$ and the height of the cone is equal to the diameter of the hemisphere, calculate the total surface area of the solid.