Class 8 Maths, Chapter 14 (NCERT). 15 questions by concept, 15 with pictures. Open “Show answer” for the answer and why.
How to teach this chapter · Common misconceptions & worked visual for this chapter
M and N are the midpoints of XY and XZ. What fraction of the area of triangle XYZ is the area of triangle XMN?
(c) 1/4
Join YN. Since N is the midpoint of XZ, triangle XYN has half the area of XYZ. Within XYN, NM halves the area again because M is the midpoint of XY. Thus XMN has one-quarter of the original area.
One rectangle has sidelengths 7 cm and 4 cm, and another has sidelengths 8 cm and 3 cm. How much more area does the first rectangle have than the second?
(b) 4 cm²
Count the unit squares: the first rectangle holds 7 times 4, that is 28, and the second holds 8 times 3, that is 24. The difference is 28 minus 24, so the first has 4 cm² more.
A bent tube of width 1 has arms 5 units and 5 units long, measured along the outside. What is the length of a straight tube of width 1 that has the same area?
(a) 9 units
The two arms share the corner square, so the bent tube covers 5 + 5 - 1 = 9 unit squares. A straight tube of width 1 with 9 unit squares is 9 units long.
Two perpendicular paths run across the full length and width of a 14 m by 12 m plot. Each path is 1 m wide. What is the total area of the crosspath?
(a) 25 m²
Add the areas of the two path rectangles. Their common square is included twice, so subtract its area once.
A square of side 6 units is cut into four pieces on a grid of unit squares. P is a square of side 3 and Q is a triangle. R is a rectangle with sides 3 units and 2 units, and S is the rest. How many of the other pieces have the same area as piece P?
(b) 1
P covers 9 unit squares. Q is half of a rectangle with sides 6 and 3, so it covers 9 as well. R covers 6 and S covers 12. So exactly one other piece, Q, has the same area as P.
A square is cut into regions 1, 2 and 3 by its diagonal and a line from the top-left vertex to the midpoint of the diagonal. Its sidelength is 4 cm, and region 3 has area 8 cm². The sidelength is doubled to 8 cm. What is the area of region 3 now?
(a) 32 cm²
Region 3 is half of the square. The square of sidelength 8 cm has area 8 times 8, that is 64 cm², so region 3 is 32 cm². Doubling the sidelength multiplies every area by 4, not by 2.
The same square is measured in inches and centimetres. Which equality correctly compares the square units?
(d) 1 in² = 6.4516 cm²
Each side of the square is 2.54 cm. Multiplying the two side lengths gives 2.54² = 6.4516 cm².
The areas and given lengths belong to rectangles A, B, C and D. What is the width of D?
(c) 2 in
B has height 21/7 = 3 in. A has height 4 + 3 = 7 in and width 28/7 = 4 in. C has width 3 + 4 = 7 in and height 35/7 = 5 in. D has height 5 + 2 = 7 in. Its width is 14/7 = 2 in.
Two rectangles each contain 24 unit squares. One has sidelengths 6 cm and 4 cm, the other 8 cm and 3 cm. By how much do their perimeters differ?
(a) 2 cm
The first rectangle has perimeter 6 + 4 + 6 + 4 = 20 cm and the second 8 + 3 + 8 + 3 = 22 cm. Their areas are the same, but their perimeters differ by 2 cm.
ABCD is a rectangle with AB = 18 cm and AD = 10 cm. AE = 10 cm and AF = 6 cm. What is the area of the hatched region FECD?
(c) 110 cm²
The rectangle has area 180 cm². Triangle AFE has area 30 cm², and triangle EBC has area 40 cm². Subtracting both gives 180 − 30 − 40 = 110 cm².
Triangle ABC has BC = 6 units, and the line from A at right angles to the line BC meets it at X, beyond B, with AX = 4 units. AC = 8 units and BY is at right angles to AC. What is the length of the altitude BY?
(b) 3 units
The area of triangle ABC is half of 6 times 4, that is 12 sq. units. The same area is half of AC times BY, so half of 8 times BY is 12 and BY is 3 units.
Triangle ABC has base BC = 4 cm. The line from A at right angles to BC meets it at E, and AE = 3 cm. What is the area of triangle ABC?
(a) 6 cm²
The area of a triangle is half of base times height. Half of 4 times 3 is 6, so the area is 6 cm².
In parallelogram PQRS, SR = 7 cm, PS = 7.6 cm and QM = 4 cm. QM is perpendicular to SR. What is the area of PQRS?
(b) 28 cm²
The base SR pairs with its perpendicular height QM. The area is 7 times 4, or 28 cm². PS is a slant side, not that height.
Rectangle ABCD has DC = 4 cm and AD = 7 cm, and the diagonal DB is drawn. What is the area of triangle DBC?
(b) 14 cm²
The diagonal divides the rectangle into two congruent triangles, so each is half of the rectangle. Half of 4 times 7 is 14 cm².
ABCD, BCEF and BFGH are identical squares, and P is where DH meets AB. The red triangle DHC has area 49 sq. units. What is the area of the blue triangle ADP?
(b) 12.25 sq. units
If each square has side s, the red triangle has base s and height 2s. Its area is s². The blue triangle has base s/2 and height s, so its area is s²/4. Therefore, the blue area is 49/4 = 12.25 sq. units.
Wrong
Multiply the base by the slanted side to find area.
Why it feels right. The slanted side looks like the second dimension.
Right
Height is the perpendicular distance to the base.
Worked example
