Mensuration

If the area of a rectangle is $120$ cm$^2$ and its length is $15$ cm, what is its width?

An equilateral triangle has a side length of $6$ cm. Find its perimeter.

A rhombus has diagonals of lengths $12$ cm and $16$ cm. Find its area.

What is the perimeter of a semicircle with a radius of $10$ cm? Use $\pi \approx 3.14$.

A chord of length $16$ cm is $6$ cm from the center of a circle. Find the area of the circle.

Find the area of a regular octagon with a side length of $2$ cm and an apothem of approximately $2.41$ cm.

The coordinates of a triangle's vertices are $(1,1)$, $(5,1)$, and $(3,7)$. Calculate the area of this triangle.

Two circles with radii $10 \text{ cm}$ and $8 \text{ cm}$ intersect such that their common chord has a length of $12 \text{ cm}$. Calculate the area of the region common to both circles.

In a circle of radius $R$, a chord $AB$ subtends an angle $\alpha$ at the center. A second chord $CD$ is parallel to $AB$ and the distance between the chords is $d$. If the area of the trapezoid $ABCD$ is maximized for a given $R$ and $\alpha$, which of the following describes the position of $CD$?

A window is in the shape of a rectangle surmounted by a semi-circle. The total perimeter of the window is $P = 10 \text{ m}$. Find the radius $r$ of the semi-circle that allows the maximum amount of light to pass through (i.e., maximizes the area).

A pyramid has a rectangular base of $10$ cm by $8$ cm and a vertical height of $15$ cm. Find the volume.

The total surface area of a cube is $216$ cm$^2$. Find its volume.

A right circular cone has a base radius $r$ and height $h=r$. If its volume is $9\pi$ cm$^3$, find $r$.

The base of a prism is a trapezium with parallel sides $8$ cm and $12$ cm, and a perpendicular distance between them of $5$ cm. If the volume of the prism is $500$ cm$^3$, find its length.

A solid right circular cylinder of radius $r$ cm and height $h$ cm is shown in the diagram. It is given that the total surface area of the cylinder is $150\pi$ cm$^2$. Find the value of $r$ that gives the maximum volume of the cylinder.

The total surface area of a solid cylinder is equal to the surface area of a sphere. If the cylinder has radius $r$ and height $2r$, find the ratio of the volume of the cylinder to the volume of the sphere.

A container is in the shape of an inverted cone of height $18$ cm and base radius $6$ cm. It is filled with water to a depth of $12$ cm. What is the volume of the water in terms of $\pi$?

A decorative garden ornament consists of a solid right circular cone of radius $r$ and height $h$ that is attached to a solid hemisphere of radius $r$. The height of the cone is exactly twice its radius ($h = 2r$). The total volume of the composite solid is $V$. If the entire surface of the solid (including the base of the cone where it joins the hemisphere) is to be painted with a uniform coat, what is the ratio of the total surface area $A$ of the composite solid to its volume $V$? Use the diagram as a guide for the shape's structure.

A right circular cone has a base radius $r$ and height $h$. A small cone of height $h/2$ is removed from the top by a plane parallel to the base. What is the ratio of the volume of the remaining frustum to the volume of the original cone?

Water flows into a conical tank at a constant rate of $2$ $\text{m}^3/\text{min}$. The tank is $6$ m deep and has a top radius of $3$ m. At what rate is the water level rising when the water depth is $2$ m?

A trapezoid is divided into a rectangle of $8 \text{ cm} \times 5 \text{ cm}$ and two identical right-angled triangles with base $2 \text{ cm}$ and height $5 \text{ cm}$. What is the total area?

A square of side $x$ has a smaller square of side $y$ removed from it. If $x = 10$ and $y = 4$, find the area of the remaining shape.

A track is composed of two parallel lines of $100 \text{ m}$ length and two semicircular ends with radius $35 \text{ m}$. Calculate the total distance of one lap. (Use $\pi = \frac{22}{7}$)

Find the area of a shape consisting of a $6 \text{ cm} \times 4 \text{ cm}$ rectangle with a right-angled triangle of base $3 \text{ cm}$ and height $4 \text{ cm}$ attached to the 4 cm side.

A solid consists of a cylinder of radius $r=3$ and height $h=5$ with a cone of the same radius and height $h=4$ removed from one end. Calculate the remaining volume.

A shape is formed by a rectangle of length $L$ and width $W$. Two identical semicircles are removed from the two width sides (the diameter of the semicircle is $W$). If $L=20$ and $W=10$, find the perimeter of the shape.

Find the area of the region in the first quadrant bounded by $y = \sqrt{x}$, the x-axis, and the line $x = 9$.

A compound shape is formed by a large circle of radius $10$ cm and a smaller circle of radius $5$ cm that is internally tangent to the large circle. A third circle of radius $r$ is drawn such that it is tangent to the large circle internally and tangent to the smaller circle externally, with its center on the line of symmetry of the first two. What is the area of the region inside the large circle but outside the two smaller circles if the third circle has radius $r = 3.33$ cm?

A swimming pool floor is shaped like a rectangle $15$ m by $10$ m, but with four quarter-circles of radius $2$ m rounded at the corners (the corners are curved inward, reducing the area). If the pool is a constant $1.5$ m deep, what is the volume of water needed to fill it? (Use $\pi \approx 3.1416$)

Calculate the perimeter of a compound shape consisting of a rectangle with dimensions $12$ cm by $8$ cm, where a semi-circle with diameter $8$ cm has been added to one of the shorter sides, and an isosceles triangle with base $8$ cm and height $3$ cm has been removed from the opposite shorter side.