Learn – Geometry – Areas

1 Quadrilaterals and triangles
- 1.1 Exercises
2 Ratios of areas to lengths
- 2.1 Exercises
  - 2.1.1 Problem set
  - 2.1.2 Problem set
3 Miscellaneous
- 3.1 Exercises

Quadrilaterals and triangles

Exercises

Problem set

In the following figure, $BD$ is a straight line segment and $AEDC$ is a parallelogram. What is the area of the parallelogram $AEDC$ ?
In the following figure, $ABCE$ is a rectangle, $ED\parallel AC$ and $DB$ is a straight line segment. What is the area of the quadrilateral $AEDC$ ?
In the following figure, $FEGH, FIJH, FKLH, FMNH$ are parallelograms. Which of the parallelograms has the smallest area?
In the following figure, $ABED$ is a parallelogram. What is the length of $AD$ ?

Problem set

In the following figure, is a straight line segment.
1. If $AB = 3$ units and $CD = 3$ units and area of $\Delta ABD = 6$ square units, what is the length of $BC$ ?
2. If $AB = 3$ units and $CD = 3$ units, what is the area of $\Delta ACD$ ?
$ABCD$ is a parallelogram with area $15$ square units. $E$ is a point on $AD$ . What is the area of $\Delta BCE$ ?
In the following figure, $BK$ is a straight line segment. Which of the following triangles is biggest in area: $ABC, ACD, ADE, AEF, AFG, AGH, AHI, AIJ, AJK$ ?
In the following figure, $AB$ and $BD$ are straight line segments. What is the area of the quadrilateral $ABDC$ ?

Problem set

In , is the mid-point of . is some point on . If area of is square units and area of is square units, calculate the area of
1. $\Delta ABD$
2. $\Delta BDE$
3. $\Delta DCE$
4. $\Delta ACE$

Problem set

In the following figure, $BC\parallel AD$ , and the area of the trapezoid $ABCD$ is $9$ square units. What is the area of $\Delta ACD$ ?
is a parallelogram and is a point lying inside the parallelogram.
1. Prove that areas of $\Delta APB$ and $\Delta CPD$ add up to half the area of the parallelogram.
2. If area of $\Delta APB$ is $3$ square units, area of $\Delta BPC$ is $4$ square units and area of $\Delta CPD$ is $7$ square units, what is the area of $\Delta DPA$ ?
$ABCD$ is a quadrilateral and $E,F,G$ and $H$ are mid-points of $AB, BC, CD$ and $DA$ respectively. Prove that the area of the quadrilateral $EFGH$ is half the area of the quadrilateral $ABCD$ .
is a quadrilateral whose diagonals and intersect at such that .
1. Prove that areas of $\Delta AOB$ and $\Delta BOC$ are equal.
2. Prove that areas of $\Delta DOA$ and $\Delta COD$ are equal.
3. If area of $\Delta AOB$ is $4$ square units and the area of $\Delta COD$ is $7$ square units, what is the area of the quadrilateral $ABCD$ ?
is a trapezoid with . The diagonals and intersect at .
1. Prove that areas of $\Delta AOD$ and $\Delta BOC$ are equal.
2. If the area of the trapezoid $ABCD$ is $32$ square units, the area of $\Delta AEB$ is $15$ square units and the area of $\Delta CED$ is $3$ square units, find the areas of $\Delta AED$ and $\Delta BEC$
$ABCD$ is a trapezoid with $AB\parallel DC$ . $E$ and $F$ are respectively the mid-points of $BC$ and $DA$ . Prove that the triangles $AED$ and $BFC$ have equal areas.

Problem set

In the following figure, $BE$ is a straight line segment, $ADEC$ is a parallelogram and $EC=CB$ . What is the area of the parallelogram $ADEC$ ?
In the following figure, $BD$ is a straight line segment and $AEDC$ is a parallelogram. What is the area of the parallelogram $AEDC$ ?
In the following figure, $ABCE$ is a rectangle, $ED\parallel AC$ and $DB$ is a straight line segment. What is the area of the quadrilateral $AEDC$ ?
In the following figure, $BD$ is a straight line segment. If $AB = 3$ units, $CD = 3$ units and area of $\Delta ABC = 1.5$ square units, what is the length of $AD$ ?

Ratios of areas to lengths

Exercises

Problem set

Two squares are such that the side of one square is double that of the other square.
1. What is ratio of their perimeters?
2. What is the ratio of their areas?
Two circles are such that the diameter of one circle is three times the diameter of the other.
1. What is ratio of their perimeters?
2. What is the ratio of their areas?
A rectangle and a square have the same area. The length of the rectangle is three times the side of the square. What is the ratio of the length to the width of the rectangle?
A rectangle and a square have the same perimeter. The width of the rectangle is half the side of the square. What is the ratio of the length to the width of the rectangle?
Base of a triangle is three times the height of the triangle. Its area is $6$ square units. What is the base of the triangle?
Base of a triangle is half the height of the triangle. Its area is $25$ square units. What is the height of the triangle?

Problem set

The perimeter of a circle is units while the area of the circle is also square units.
1. What is the value of $k$ ?
2. What is the radius of the circle?
The perimeter of a circle is units while the area of the circle is square units.
1. What is the value of $k$ ?
2. What is the radius of the circle?
Length of a rectangle is seven times its width. Its perimeter is $96$ units. What is its length?
Ratio of length to width of a rectangle is $9:5$ . Area of rectangle is $405$ square units. What is its perimeter?
If the side of a square is doubled, what is the ratio of the area of the original square to the area of the new square?
Two circles are such that the circumference of first one is one-third the circumference of the second one. What fraction of the area of the second circle is the area of the first circle?

Miscellaneous

Exercises

Problem set

In the following figure, $ABDC$ is a rectangle. What is the area of trapezoid $ACDG$ ?
Sides of a pentagon are in the ratio $6:8:10:11:12$ . Its perimeter is $23.5$ units. What is the median of the sides of the pentagon?
An isosceles right triangle has area $4$ square units. What is its perimeter?

Problem set

What is the area of the following triangle?
In the following figure, $ABED$ is a parallelogram. What is its area?
In the following figure, $ABED$ is a parallelogram. What is its area?
In the following figure, $ABED$ is a parallelogram. What is its area?
In the following figure, $ABED$ is a parallelogram. What is its area?

Problem set

What is the area of a regular hexagon whose side is $2$ units?
In the following figure, $DF$ is a straight line segment, $AB \parallel FD$ and $AC\parallel BD$ . What is the area of the quadrilateral $ABDC$ ?
In the following figure, $ABED$ is a parallelogram. What is the length of $AD$ ?
In the following figure, $ABED$ is a parallelogram. What is its area?
In the following figure, $MQ\parallel US$ , $MZ\parallel NS$ , $PU\parallel QV$ , $UV=ZS$ and $\angle USQ = 90^\circ$ . What is the area of the portion of the figure that is colored gray?

Problem set

In the following figure, $ACFE$ and $ADCB$ are two squares. And, $B$ is the center of the square $ACFE$ . What is the ratio of area of square $ACFE$ to the area of square $ADCB$ ?
In the following figure, $ABCDEF$ is a regular hexagon. What is the area of the quadrilateral $ABCD$ ?
In the following figure, $ABCDEF$ is a regular hexagon. $G$ and $H$ are midpoints of $ED$ and $AB$ respectively. What is the area of the polygon $AHGEF$ ?
Following figure shows a circle and a square inscribed in the circle. What is the area of the shaded portion?
Following figure shows a square and a circle inscribed in the square. What is the area of the shaded portion?