📐 geometry

1. **State the problem:** Find the volume of the right pentagonal prism with base height 11 mm, base width 8 mm, and prism length 15 mm. 2. **Formula:** The volume $V$ of a prism i

1. **Stating the problem:** We are given a right prism with volume 480 cm³. The base ABCD is a quadrilateral where AD is parallel to BC. Given \(\angle CDA = 90^\circ\), \(BC = 4\)

1. **Problem statement:** A cylindrical block of metal with diameter 1 m and height 2 m is melted to form 2000 cubes. Find the length of each side of the cube in cm. 2. **Formula a

1. **Stating the problem:** A cylindrical block of metal with diameter 1 m and height 2 m is melted to form 2000 cubes. We need to find the length of each side of the cube in cm.

1. **Problem statement:** We have a circle with points L, N, Q on the circumference, where LN = NQ, and RA is tangent at L. Given \(\angle MLN = 63^\circ\), we need to explain why

1. **Problem Statement:** We have a circle with center O and points P, Q, L, N on the circumference. Given that LN = NQ and RM is a tangent at L. Angle MLN = 63°.

1. **State the problem:** Find the values of angles a, d, and e in the circle with center O, points L, M, N, R on the circumference, tangent PQ at R, given \(\angle PRL = 48^\circ\

1. **Problem statement:** We have two similar shapes, S and T. Shape S has a perimeter of 12 cm and a height of 4 cm. Shape T has a height of 40 cm. We need to find the perimeter o

1. **Problem Statement:** We need to find:

1. **State the problem:** We need to find the area of a right triangle with legs measuring 6 cm and 7 cm. 2. **Formula:** The area $A$ of a triangle is given by

1. **State the problem:** We need to find the area of the given irregular shape composed of a rectangle and a right triangle. 2. **Identify the shapes and dimensions:**

1. **State the problem:** We are given the volume of a cone as $378\pi$ cm$^3$ and the radius $r=9$ cm. We need to find the missing height $h$. 2. **Formula:** The volume $V$ of a

1. **State the problem:** We need to find the radius $r$ of a cone given its volume and height. 2. **Given formula:** The volume $V$ of a cone is given by

1. **State the problem:** We have a triangle with sides 37 mm, 35 mm, and 38 mm, and one angle of 76°. We need to find the size of angle $\theta$ to 2 decimal places.

1. **State the problem:** We have three points A, B, and C. The bearing of B from A is 050° and the bearing of C from B is 110°. We need to find the size of angle ABC. 2. **Underst

1. **State the problem:** We need to determine the maximum occupancy for each room based on the area and the occupancy limit of 1 person per 1.7 square meters. 2. **Calculate the a

1. The problem asks for the formula to find the volume $V$ of a composite solid consisting of a hemisphere on top of a cone. 2. The volume of a hemisphere is given by the formula:

1. **Problem statement:** Given a circle with points A, B, C, and F on the circumference, and a tangent DE touching the circle at point B. The angle at F is 96° and the angle betwe

1. **Problem Statement:** We have two cylinders:

1. **State the problem:** Determine if polygons EFGH and WVUT are similar based on their given angles and side lengths. 2. **Recall similarity criteria for polygons:** Polygons are

1. **State the problem:** The user wants to solve for the values of $x$, $y$, and $z$ related to angles formed by chords intersecting inside a circle with a given 50° angle at the