Maths circle: Areas – We Solve Problems

Maths circle: Areas

Today we will solve some problems about finding areas of geometric figures. All you need to know to solve every problem in this set is: How to calculate the area of a triangle? ($\frac12 a \times h$), rectangle? ($a \times b$), circle? ($\pi r^2$)

Problems:

1.

Area , Area of a trapezium

In a trapezium $ABCD$, the side $AB$ is parallel to the side $CD$. Show that the areas of triangles $\triangle ABC$ and $\triangle ABD$ are equal.

2.

Area , Triangle area

The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is $18$, find the area of the small blue triangle.

3.

Area , Area of a circle, sector, segment

On the left there is a circle inscribed in a square of side 1. On the right there are 16 smaller, identical circles, which all together fit inside a square of side 1. Which area is greater, the yellow or the blue one?

4.

Area , Area (other)

In a pentagon $ABCDE$, diagonal $AD$ is parallel to the side $BC$ and the diagonal $CE$ is parallel to the side $AB$. Show that the areas of the triangles $\triangle ABE$ and $\triangle BCD$ are the same.

5.

Area , Triangle area

Which triangle has the largest area? The dots form a regular grid.

6.

Area , Area of a circle, sector, segment , Circles

What is the ratio between the red and blue area? All shapes are semicircles.

7.

Area , Area of a parallelogram

In a parallelogram $ABCD$, point $E$ belongs to the side $CD$ and point $F$ belongs to the side $BC$. Show that the total red area is the same as the total blue area:

 

8.

Area , Area (other) , Area of a polygon

The figure below is a regular pentagram. What is larger, the black area or the blue area?

9.

Area , Area (other) , Area of a circle, sector, segment

A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?

10.

Area , Area of quadrilateral

In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length $10$.

11.

Area , Area of a parallelogram

In a parallelogram $ABCD$, point $E$ belongs to the side $AB$, point $F$ belongs to the side $CD$ and point $G$ belongs to the side $AD$. What is more, the marked red segments $AE$ and $CF$ have equal lengths. Prove that the total grey area is equal to the total black area.

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