Practice (54)
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We are asked to find the area of the shaded (grey) region where the distance between each dot is 1 cm.
In trapezoid $ABCD$, $AB = BC = 2AD$ and $AD= 5$. We are asked to find the area of trapezoid $ABCD$.
Circle O has diameter AE and AE = 8. Point C is on the circumference of the circle such that segments AC and CE are congruent. Segment AC is a diameter of semicircle ABC and segment CE is a diameter semicircle CDE. What is the total combined area of the shaded regions?
The cube shown has a side length of $s$. Points $A$, $B$, $C$ and $D$ are vertices of the cube. We need to find the area of rectangle $ABCD$.
The diagram shows 8 congruent squares inside a circle. Find the ratio of the shaded area to the area of the circle.
In triangle $ABC$, where $AC$ > $AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are 5 and 6, respectively, compute the area of triangle $ABC$.
Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand?
Quadrilateral $ABCD$ satisfies $AB = 4$, $BC = 5$, $DA = 4$, $\angle DAB = 60^{\circ}$, and $\angle ABC = 150^{\circ}$. Find the area of $ABCD$.
A semicircle with diameter $AB$ is constructed on the outside of rectangle $ABCD$ and has an arc length equal to the length of $BC$. Compute the ratio of the area of the rectangle to the area of the semicircle.
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $AB$ = 4, $CD$ = 8, $BC$ = 5, and $AD$ = 6. Given that point $E$ is on segment $CD$ and that $AE$ is parallel to $BC$, find the ratio between the area of trapezoid $ABCD$ and the area of triangle $ABE$.
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
Given a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?
In trapezoid $ABCD$ with bases $AB$ and $CD$, we have $AB = 52$, $BC = 12$, $CD = 39$, and $DA = 5$ (diagram not to scale). The area of $ABCD$ is
The vertex $E$ of square $EFGH$ is at the center of square $ABC$D. The length of a side of $ABCD$ is 1 and the length of a side of $EFGH$ is 2. Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J$. If angle $EID = 60^\circ$, what is the area of quadrilateral $EIDJ$?
Let $a, b, c$ be respectively the lengths of three sides of a triangle, and $r$ be the triangle's inradius. Show that $$r = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(b+a-c)}{a+b+c}}$$
Let $a, b, c$ be respectively the lengths of three sides of a triangle, and $R$ be the triangle's circumradius. Show that $$R = \frac{abc}{\sqrt{(a+b+c)(b+c-a)(c+a-b)(b+a-c)}}$$
A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
If the area of a circle's inscribed square is 60, what is the area of its circumscribed square?
What is the area of a trapezoid the lengths of whose bases are 10 and 16, and the lengths of whose legs are 8 and 10?
A diagonal of a square intersects a segment that connects one vertex of the square to the midpoint of an opposite side, as shown. If the length of the shorter section of the diagonal is 2, what is the area of the square?
Find the area of shaded area if the side length of the square is 1.
In triangle $ABC$ , side $AC$ and the perpendicular bisector of $BC$ meet in point $D$, and $BD$ bisects $\angle ABC$. If $AD=9$ and $DC=7$, what is the area of triangle ABD?