📐 geometry

1. Let's clarify what "center" means in different contexts, such as the center of a circle, center of a polygon, or center of mass. 2. For a circle, the center is the point equidis

1. **State the problem:** We need to find the length of side $c$ in a triangle where two sides are $3$ and $10$, and the included angle between them is $120^\circ$. 2. **Formula:**

1. **State the problem:** We need to find the volume $V$ of a sphere given the diameter is $18$ cm. 2. **Recall the formula for the volume of a sphere:**

1. **State the problem:** We need to find the area of a circle with radius $5$ km. 2. **Formula for the area of a circle:** The area $A$ of a circle is given by the formula:

1. **State the problem:** We need to calculate the circumference of a circle with radius $42$ mm, and express the answer in terms of $\pi$. 2. **Formula for circumference:** The ci

1. **Problem 1:** Find the volume of a sphere with radius $r=3$ cm. 2. **Formula:** The volume $V$ of a sphere is given by

1. **Problem Statement:** We need to find the size of the angle $BEC$ in the cuboid. 2. **Understanding the cuboid and points:** The cuboid has vertices $A$, $B$, $C$, $D$ on the t

1. **Problem statement:** We have a right triangle with sides 3, 4, and 5, where $AB=5$, $AC=4$, and $BC=3$. The triangle is folded so that point $A$ falls on point $B$. We need to

1. **Problem statement:** Given an isosceles triangle $ABC$ with $AB=AC$ and area 40, inside which there are 7 smallest similar triangles each with area 1, find the area of trapezo

1. **State the problem:** We need to find the total area of a sign composed of a semicircle on top of an isosceles triangle. The semicircle has a diameter of $48$ cm, and the trian

1. **State the problem:** We need to find the total area of a sign composed of a semicircle and an isosceles triangle. The semicircle has a diameter of $48$ cm, and the triangle ha

1. **State the problem:** We are given the area of a circle as $83$ cm$^2$ and need to find its radius rounded to $1$ decimal place. 2. **Formula:** The area $A$ of a circle is giv

1. **State the problem:** We need to find the area of a compound shape made of a semicircle and a rectangle. 2. **Identify dimensions:** The rectangle has width $3$ cm and height $

1. **State the problem:** We need to find the area of an annulus (ring-shaped figure) formed by cutting a smaller circle of radius $5$ cm from the center of a larger circle of radi

1. **State the problem:** We need to find the area of a scalene triangle with a base of $7$ cm and a height of $6$ cm. 2. **Formula for the area of a triangle:**

1. **State the problem:** We have a rectangle and a triangle with the same area. The rectangle has height $8$ cm and width $5$ cm. The triangle has base $d$ cm and height $5$ cm. W

1. **Problem Statement:** We have two intersecting lines $l_1$ and $l_2$ with points marked on them. We want to find:

1. **Problem Statement:** We have a parallelogram ABCD with points E on BC and K on CD such that $\frac{BE}{EC} = 3$ and $\frac{CK}{KD} = 2$. We want to find the ratio of the area

1. **State the problem:** We have a square-based pyramid with a base edge of $16$ mm and triangular faces with a slant height of $15$ mm. Jade has paint that covers $5580$ mm². We

1. **State the problem:** Calculate the surface area of a prism with trapezium end faces and rectangular/parallelogram side faces. 2. **Identify the dimensions:** The trapezium has

1. **State the problem:** Calculate the surface area of the given triangular prism with edges 4 cm, 8 cm, 13 cm, 7 cm, 12 cm, and 18 cm. 2. **Identify the shape and dimensions:** T